brychkov-handbook of special functions-derivatives, integrals, series, and other formulas
TRANSCRIPT
Special Functions: Derivatives, Integrals, Series and Other
Formulasand Other Formulas
H A N D B O O K O F
Yury A. Brychkov Computing Center of the Russian
Academy of Sciences Moscow, Russia
Special Functions Derivatives, Integrals, Series
and Other Formulas
H A N D B O O K O F
Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742
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Contents
+**,!"! '
+**,!"!" '
1.4. The Sine si (z) and Cosine ci (z) Integrals ( # +**,!"! (
1.5. The Error Functions erf (z) and erfc (z) -
' +**,!"! -
& +**,!"! .
( +**,!"! .
- +**,!"! - +**,!"!"
/ +**,!"!
. +**,!"! '
+**,!"! /
1.12. The Hankel Functions H (1) ν (z) and H
(2) ν (z)
+**,!"!
# +**,!"! & # +**,!"!0 #.
1.16. The Anger Jν (z) and Weber Eν(z) Functions # '
& +**,!"! #'
& +**,!"!0 #(
( +**,!"! #-
- +**,!"! '&
/ +**,!"! '/
. +**,!"! &
+**,!"! &
+**,!"!" &#
+**,!"! &# +**,!"!" &&
+**,!"! && +**,!"" &/
# +**,!"! &/
.
' +**,!"! (.
' +**,!"" (
1F1(a; b; z) (
1.28. The Whittaker Functions Mµ,ν(z) and Wµ,ν(z) -.
- +**,!"! -.
/ +**,!"! -.
/ +**,!"" -'
. +**,!"! -& . +**,!"" -(
4.3. The Sine Si (z) and Cosine ci (z) Integrals -
# )0 - # )0 - # )0!! -
#' )"00 -#
4.4. The Error Functions erf (z), erfi (z) and erfc (z) -#
## )0 -# ## )0!" -'
4.5. The Fresnel Integrals S(z) and C(z) -/
#' ) 0 -/
#' ) 0 /.
#& )0 /
#& )0!" /
#& )0 /
1
#( )0 /& #( )0!" /( #( )0 /( #(# )0!! //
#- )0 .# #- )0 .#
#/ ) 0 .'
#. ) 0 & #. )"0 &
# )H L
0 ( # )H
0*
4.12. The Kelvin Functions berν(z), beiν (z), kerν(z) and keiν (z)
# ) 0
# )"00
## )0
#' )0 # #' )0 & #' )0" #
#& )0 # #& )0 #
#& )0 #& #&# )"0 #&
#( )0 #( #( )0!" #-
'# #- )
'( #/ )
&
# )K 0 &'
# )K !"!%"0 &(
# )K 0!! &-
## )K 0* (
#' )K 0 (#
# )E 0 (/
# )E !"!%"0 -
# )E 0!! -&
## )E 0* -/
#' )E 0 /
# )D 0% ..
## )0 .
' ./ ' "0 '
5.2. The Incomplete Gamma Functions γ(ν, z) and Γ(ν, z)
' ' "0 '
' # ' "0 #
'# ' '# "00 ' '# "0 (
'' - '' 0" - '' "0 /
'& H 0L
'( . '( 0" ' '( "0 ' '(# & '(' - '(& "0 -
'- / '- "0 '- '-# '-' "0 ' '-& '
'/ ( '/ 0" #. '/ "0 # '/# # '/' # '/& "0 ##
## '.
##
'.
#& '.
#-
'.#
& '
& '
&' '
&'
'#
/ '
!
/ '
!
/
' !
'
#
1F1(a; b; z) #'# # '# 0" # '# "0 #
5.15. The Tricomi Confluent Hypergeometric Function Ψ(a; b; z) # '' # '' 0" ##
5.16. The Gauss Hypergeometric Function 2F1(a, b; c; z) ##'& ##
'( #- '( 0" # '( # '(# 0" ##'(' "0 #' '(& #'
5.18. Multiple Sums # &
& # & 0 ##( & "0 ##( &# ##/
& #'
&# # #' &# # #'
&' # #0 #'
&' # #0 #''
&' # #0 # #'-
&& #'/
&( 0% #&.
&- ! #&'
&/ #('
&. H 0L
& #-& & 0 #-& & "0 #--
& #-/
/'
&#
#/'
&#
//
&'
#//
&'
'.'
&& !
'.'
&& !
&( '.-
8.1. The Hypergeometric Functions '&
m,n p,q
' - % &' - " &'' &&/ )0 0 &( )0% &(/
1
Preface
1.1. Elementary Functions
1.1.1. General formulas
1. ! '#" "$&% ')(* $ ! % ,+ &- . ¤ 2. 0/ 0 &-21 43 ¢ '65 $ (87! %$9% ')( $ (:7! % £ / -;&- <+ &- . / = '8>?7A@CB 3. ED EF 7GIH 3 ¢ 3 ¢ &-21 KJ ' $L "" '')( $ (:7! % ,+ &- . F 7G = '8>?7A@CB 4. EDM &-21 EF 7 GIH 3 ¢ -;&-21ON F 7 G < PQ , & N & ¤ 5. 1 &R 0& ¤¤4¤ ST0& U4 J '$ LV + U4.1 & &- UXW4 J ')( $ $ LV + XW4. 0& ¤¤4¤
Y &- U -<Z Z Z - X[<\]W [<\#U ^ ')( $ (_BB B<( $ ( " $ ( " ` + [<\#U .S -21 &R + &- U -<Z Z Z - X[<\#U4.S & ¤ 6. EDM SEa F 7 G S D S -21Ab F 7 G9HOH 43 ¢ SEa -;&-21 N F 7 G
&9 S b 0& N 0& ¤ 1.1.2. Algebraic functions
, 5. c f & c F m G ( &! '
J 1 ( L ' c -; 0f & c -; c -; a 1 F ¢ m G ¤ 6.
3 ¢ 0 fI& c -; - c F O 5 m 5 m G ¤ 7. -21 0f & -21
£ 3 ¢ a 1 f -21 f & -;&-21 ¢ 3 F m 5 G + a 1 . a 1 F m 5nO m 5 G! ¤ 8. - c -21 ] fg3h& c
£ - '! % J 1 ( L ' - c -21 0fg3h& c -; c -; a 1 F#" m G ¤
9. c fg3h& &- c -21 43 ¢ f c -; 0fg3h& - c -21$ - c F 5 m m G ¤ 10. &- c -21 0f & c
F3 7 G (9! ' J 1 (% L ' &- c -21 f & c -; c -; a 1 F ¢ m8G ¤
11. - c -!&4 0fg3h& c 3 ' % J & L 'J ('6( 1 L '%& f -21 O - c -21 fg3h& c -; c -; a 1 a 1 F#" m G ¤
12. &-21 0fg3h& c J 1 L 'J 1 ( L ' f -21 0fg3h& c -; c -; a 1 F " m G ¤ 13. a 1 0f & c
,
19. 0f & -;&-!&4 f -21 ] f- & -;&-21Oj a 1 )+* mm 5 , ¤ 20. 0f & f j F ¢ m G ¤ 21. -;&-21 0f & 3 ¢ -;&-21 j F ¢ m G ¤ 22. &-21 fh& -;&-21 43 ¢ -21 0f h& -;&-21 j )+* m 5: , ¤ 23. &-21 0f & 3PfI -21 j F" m G ¤ 24. a 1 0f & a 1 J & L ''g5 7 f 1 0f & 1 F ¢ m G ¤ 25.
J & L '#'Q5_7! f a 1 1 a 1 F" ¢ m G ¤ 26. &-21 0f & a 1 F 7 G f a 1 -21 a 1 F#" ¢ m G ¤ 27. a 1 0f & &-21 F 7 G f 1 0f & -21 F " ¢ m G ¤ 28. a 1 0fg3h& &-21 43 ¢ F 7G f a 1 fg3h& -21 a 1 F#" m G ¤ 29. &-21 0f & &-21 F 7G f -21 f- & -21 F#" ¢ m G ¤ 30. &-21 0f & -; F 7G -21 f & -; )+* mm 5 , ¤ 31. a 1 0f & -;&-21 F 7 G 1 f- & -;&-21 )+* mm 5n , ¤ 32.
43 ¢ F 7G f- & -;&-21 a 1 ) * m 5 , ¤ 33. &-21 0f & -;&-21
F 7G fI& -21 0f & -;&-21 a 1 ) * mm 5 , ¤ 34.
43 ¢ F 7 G -21 0fh -;&-21 )+* m 5: , ¤ 35. a 1 0f & -;&-
43 ¢ £ - O'65_7X! %'65 7! % 1 f- & -;&- F ( m 5 m G ¤ 3
3 36.
J & L '#'Q5_7! F m G 1 f & -;&-!&4 a 1 )+* mm 5n , ¤ 37. 0f9 3h c 3 £ fI ('9! '' ( &! ' 0f9 3h c -; c -; a 1 F m G ¤ 38. 0f9 3h -21
f -21 fg3h& -;&-21 ¢ 3 F E( m 5 m G + a 1 . a 1 F ( m G ¤ 39. 0f9 3h -21 £ f -21 f9 3h O -;&-21 a 1 F m m (8 G ¤ 40. a 1 0f9 3h -21 £ ¢ f -21 ; f9 3h O -;&-!&4 a 1 F m m (8 G ¤ 41. -21 0f & -21
43 ¢ - + a 1 . 0f & - + a 1 . j F 5 m m 5 G ¤ 42. 0f9 -21 R 43 ¢ f9 - + a 1 . j F m 5 G ¤ 43. 0f9 3h 43 £ fI j F m G ¤ 44. 0f9 3h &-21 R 43 £ fI F 7 G f9 3h -21 F m G ¤ 45. 0fIQ3h &-21 R 43fI F 7 G 0fIQ3h -21 F" m G ¤ 46. 0f 3h a 1 ( m ! ''65_7 F G 0f 3h 1 F m G ¤ 47. -;&-21 0f9 3h 43 £ fI -;&-21 j F m G ¤ 48. &- c -21 0 3hf9 c
£ fI ('9! '')( &! ' &- c -21 0 3hf c -; c -; a 1 F m G ¤ 49. 7 5 m F $ f G 1
3 ¢ m F 7 G 0 f9 - + a 1 . F O'65_7 m G ¤
1.1.3. The exponential function
1. k c - c -; - c -; 0fI& ¤ 2. c -R 43 ¢ c -; -R - c -21 F m G ¤
5
3. ED W H 3A f W R/ f& ¤ 4. EDM &-21 R W H f -;&-21 R W F m G ¤ 5. ED - W H 3 ¢ f - W / f& ¤ 6. EDM &-21 #-R W H f -;&-21 #-R W F m G ¤ 7. / 8F m G a 1 + 1A- . &-21 f;/ 1 -; 0f;/ ¤ 8. D 7 H / *F m G a 1 - + 1 a . a 1 f;/ -;&-21 f;/ 0 ¤ 9. k &-21 R
3 ¢ / *F mG a 1 - + a . D &-21 F m G 1 -; F m GIH ¤ 10. k &-21 R
43 ¢ / F m G a 1 - + & a . D a 1 F m G -;&-21 F m GIH ¤ 11. - 3 ¢ m ' & ']" I+ 1A- . &-21 f / ¤ 12. D 7 - H 43 ¢ m '%& '#" - + 1 a . a 1 0f / ¤ 13. &-21 -R m '%& ']" - + a . &-21 F m G ¤ 14. k -;&-21 #-R m '%& ']" - + & a . a 1 F m G ¤ 15. D + -21 . U H 3 ¢ a 1 R/ *F m G a 1 + 1A- . + .1 -; f;/ = 7@CB 16.
43 ¢ / 8F m G a 1 + 1A- . + .&-21 0f;/ = 7]X@CB 17. D 7 + -21 . U H
3 ¢ a a 1 R/ *F m G a 1 - + a 1 . + . a 1 0f;/ = 7@CB 18.
/ F m G a 1 - + a 1 . + .-;&-21 0f / = 7]X@CB ;
5 1.1.4. Hyperbolic functions
1. &c fI&0 ' % &c -; c -; 43fI&<3 #- c -; fI& ¤ 2. c fI&0 ' % c -; c -; 43fI& - c -; fI& ¤ 3. fI&0
£ 3fI ")" $ 5 7! " " 5_7! 3 "'& 8 S 43 ¢ S J $ L ¤
#$ ¤
#$ ¤
12. D & H £ 43 ¢ F 5 G a 1
:
13. ED H 43 ¢ a 1 ' % '%&
# ( 7! ' ' % ' & a & N a 7( &B B B ( &B B B 7 ( 9BB B #7( #7,5 &B B B #7<5 ! 7 #$ ¤
14. D &-21 F m GIH 3 ¢ / F m G a 1 - + a . 1 -; F m G ¤ 15. ED &-21 F m GIH 3 ¢ / *F m G a 1 - + a . &-21 F m G ¤ 16. D &-21 F m GIH 3 ¢ / F m G a 1 - + & a . a 1 F m G ¤ 17. ED &-21 F m GIH 3 ¢ / F m G a 1 - + & a . -;&-21 F m G ¤ 18. 0f / '%& f a 1 I+ 1A- . &-21 '65 $ !$&% ')( $ ! 43fI - - -; a 1 0f / = '*>?7R@CB
19. 0f / ' & f a 1 I+ 1A- . &-21 '65 $ !$9% ')( $ ! 43fI - - &- -21 f / = '*>?7R@CB
20. m !
' & f a 1 - + a 1 . &-21 '65 $ !$9% ')( $ ! 3PfI - - &- a 1 f / = '*>?7R@CB 21. m !
'%& f a 1 - + a 1 . &-21 '65 $ !$&% ' ( $ ! 43fI - - -;&-21 0f / = '*>?7R@CB
1.1.5. Trigonometric functions
1. 0fI& f F fI ' G ¤ <
; 2. 0fI& f F fI ' G ¤ 3. 0fI& 43 ¢ + a 1 . f 1
( 7! " " J '65_7$ 5 7 L a 1 fI& Y S J $ L 3 £ D 3 £ fIQ3 7( ( 7! ' H = = @ !C@CB
4. 3 ¢ + a 1 . f 1
( 7X! " " J '65_7$ 5_7 L S J $ L 3 £ SEa 1 fI& Y + - . - S
3 ¢ J $ (* 5 L a fI& '
! = @O !#B 5.
3 ¢ + a 1 . f fI& 7 " + - . S 43 ¢ S J $ 5 L Y SEa 0fI&
43 ¢ J $ L £L ¢ '
! = @ !#B 6.
£ AfI " $ 5_7X! 5 7! 3 "'& 8 S 3 ¢ S J $ L = = @ !C@CB
7. 3 £ a 1 m ' ' % m 5 ! '%& m ' ! '%& D + . F 5n m G 3 + . F 5n m G9H
3 43 ¢ m ' ! '%& D + . F (* m G 3 + . F 3 5n m GIH ¤ 8. fI&0
3 ¢ f 1 ( 7! " " J '65_7$ 5_7 L S J $ L 3 £ SEa 1 fI&
Y + - . - S 3 ¢ J $ (* 5 L a 0fI&
' ! = @ C7X O! B
7 " S 43 ¢ S J $ L £ ¢ Y + - .
3 ¢ J $P5 L a 0fI& '
! = @ C7 O! B L
10.
43 ¢ a 1 ' %m ' & m ' ! '%& D + . F 5 m G 3 + . F m G9H 3 43 ¢ m ' ! ' & D + . F ( m G 3 + . F3 m G9H ¤
11. 0fI& 43 ¢ a 1 £ fI 1 $"'& m ]! "
Y D 3 ¢ 4fI$ 7( ( 7! ' H S 1 3 ¢ S J $ L = = @ C7 !C@CB 12.
43 ¢ 4 a 1 £ fI fI& 1 7 " S 1 3 ¢ S J $ L
Y + a V- . 3 ¢ J $ (87Q( Q5_7 L - a 1 fI&
' ! = @ ! B
13. m '
' & D + . F 7 m G 3 3 ¢ + . F 7 3 m GIH ¤ 14. fI&0 43 ¢ + &-21 . C £ fI fI& 7 " S 1 3 ¢ S J $ L
Y + a V- . 43 ¢ J $ (:7Q( 5_7 L - a 1 fI&
' ! = @ ! B
15. m '
' & D 3 ¢ + . F]3 m G 3 + . F ¢ m G9H ¤ 16. c fI&0 ' % c -;
c -; 43 AfI&3 - c -; AfI& ¤
17. c 0fI& ' % c -; c -; 43 AfI& - c -; AfI& ¤
18. fI 0 ( 7X! ' f + &- . D W & / fE3 - W / f H ¤
19. 0fI ( 7X! ' f D W & / f - W / f H ¤
20. ED &-21 m H 3PfI -;&-21 F m ' G ¤ 21. D &-21 m H 3PfI -;&-21 F m ' G ¤
P
; 22. f / 0 / *F m G a 1 I+ 1A- . 1 -; f / ¤ 23. 0f;/ 3 ¢ / *F m G a 1 I+ 1A- . &-21 0f;/ ¤ 24. S 0f;/ <
#$ ¤
#$ ¤
30. ED H £ #F ( G a 1
Y a & N a 7 1 ( 9BB B 1 ( 1 5 &B B B 1 5 & ( &B B B & ( & 5 &B BB & 5 ! 7 #$ ¤
31. D H £ 43 ¢ -;&-21 Y a & N a 7 V( 9BB B V( &B B B ! 77 ( &B BB #7( #7<5 &B B B V7<5
#$ 3 43 ¢ ' % '%& ¤ N
32. ED &-21 m H 43 ¢ / 8F mG a 1 - + a . 1 -; F m G ¤ 33. ED &-21 m H / 8F mG a 1 - + a & . a 1 F m G ¤ 34. D &-21 m H / F m G a 1 - + a . &-21 F m G ¤ 35. ED &-21 m H 43 ¢ / 8F mG a 1 - + a & . -;&-21 F m G ¤ 36. f / 0 £ - &-21 O/ f a 1 I+ 1A- .
Y &-21 43fI - '65 $ !$9% ')( $ ! - -; a 1 f / = '8>?7A@ B 37. 0f / 3 ¢ £ - &-21 / f a 1 I+ 1A- .
Y &-21 f - '65 $ !$9% ')( $ ! - &- -21 f / = '8>?7A@CB 38. m !
43 ¢ £ - &-21 / f a 1 O - + 1 a .
Y &-21 f - '65 $ !$9% ')( $ ! - &- a 1 f / = '8>?7A@CB 39. m !
£ - &-21 / f a 1 - + 1 a .
Y &-21 43 ¢ f - '65 $ !$&% ')( $ ! - -;&-21 0f / = '8>?7A@ B 40. RED
H 3# ¤
42. R a D H 643 ¢ £ a 1
¤ 43. R a & D
H 3V ¤
47. R a & D H 3 ¢ a 1 £ a 1 & ¤
48. f;/ 0f;/ f;/ ¢ / 8F m G a 1 I+ 1A- . Y D O'65 ! <1 f / £ '
'Q5 ! <1 f / £ H ¤ 49. f;/ 0f;/
f;/ ¢ a 1 / *F m G a 1 I+ 1A- . Y D O'65 ! 2 1 f / £ K3 O'65 ! 2 1 f / £ H ¤
50. 7 f;/ 0f;/ f;/ ¢ / *F m G a 1 - + a 1 . Y D O'65 ! 2<1 f / £ '
'Q5 ! 2<1 f / £ H ¤ 51. 7 0f / f;/
f;/ ¢ a 1 / *F m G a 1 - + a 1 . Y D O'65 ! 1 f / £ E3
'Q5 ! 1 f / £ H ¤ 1.1.6. The logarithmic function
1. / / f 43 ¢ &-21 ')(:7X! % -; f9 -; j &-21 ) 5 m 5 m , = '*>?7R@CB
2. k &-21 / f / f ')(87! %O 3 ')(87! %O m 5 m j &-21 F 5n m m 5 m G = '*>?7R@CB
3. D F $ f G9H 43 ¢ &-21 3 ¢ 0 f9 O -; j &-21 F 5 m G = '*>?7R@CB
4. D &-21 F f f GIH ')(87! % 3 ')(:7X! % f f9 -; j &-21 F m 5 m G = '*>?7R@CB
5. ED m 5 m (8<H £ £ 3 ¢ ;0f9 3h -;&-21 &-21 F m m (* G = '8>?7A@CB !,
6. a 1;D m 5 m (8<H £ £ 0f9 3h O -;&-21 a 1 F m m (8 G ¤ 7. D m 5 m ( H
3 ¢ f &-21 1 -; f 3h& -; j + 1 -;9lC-; .&-21 F ¢ 3 m G = '*>?7R@CB 8. D &-21 m 5 m ( H
43 ¢ 3 ¢ fI &-!&4 ] Q3hf9 O -; j + 1 -;9lC-; .&-21 ) ¢ 3 m , = '*>?7R@CB 9. EDM 1 f9 3h& EDM &-21 m 5 m ( H H ¡ = '8>?7A@CB 10. ED &-21 0f9 3h& ED m 5 m ( HOH ¡ = '8>?7A@CB 11. ED -21 0f9 3h& ED &-21 m 5 m ( HOH
£ ) 3 m , -;&-21 m 5 m ( ¤ 12. D a 1 0f 3h& D -21 m 5 m ( HOH '! % ' m 5 m ( ¤ 13. ED a 1 PED -21 0f 3h& &-21 m 5 m ( HOH
43 ¢ F 7 G f f 3h& -;&-21 m 5 m ( ¤ 14. ED &-21 PED f 3h& &-21 m 5 m ( HOH
43 ¢ F 7 G f -21 f 3h& -;&-21 m 5 m ( ¤ 15. &-21 ¢ 3hfI& ¢ 3hfI&0 ¡ = '8>?7A@CB 16. ¢ 3hfI& &-21 ¢ 3hfI&0 ¡ = '8>?7A@CB 17. a 1 ¢ 3hfI -21 -21 ¢ 3hfI ¢ 3hfI& 43 ¢ f ¢ 3hfI& -;&-21 ¢ 3hfI& ¤ 18. a 1 ¢ 3hfI -21 ¢ 3hfI&0 4 f ¢ 3hfI& ¤ 1.1.7. Inverse trigonometric functions
1. fI&0 3A &-21 3 ¢ f ¢ 3hf9 -; j &-21 F m 7( m G= '*>?7R@CB 3
< 2. 0fI&
3 ¢ &-21 3 ¢ f ¢ 3hf9 -; j &-21 F m 7 ( m G = '*>?7R@CB 3. fI&0
43 ¢ £ 3 ¢ f9 a 1 ; ¢ f9 O -;&-21 &-21 F 7 7<5 m G = '*>?7R@CB 4. a 1 fI&0
43 ¢ £ f9 a 1 ¢ f9 O -;&-21 a 1 F 7 7<5 m G ¤ 5. fI&0 43 ¢ a 1 £ 3 ¢ f9 a 1 ; ¢ f9 O -;&-21 &-21 F 7 7<5 m G = '*>h7A@CB
6. a 1 fI&0 43 ¢ a 1 £ f9 a 1 ¢ f9 O -;&-21 a 1 F 7 7<5 m G ¤
7. f;/ 0 ( ! ']" 3 ¢ f 0Q3hf -; j &-21 ) 7( m m m (8 , = '*>?7R@CB
8. f;/ 0 ')(:7X! % fI 1 -; f $ ¢ -; j + 1 -;9lC-; .&-21 £ f ¢ = '*>?7R@CB
9. EDM &-21 m H 3 '#" 3 ¢ f -21 0Q3hf9 -; j &-21 ) ( m m m (8 , = '*>?7R@CB
10. D &-21 m H ( 7! ' 3 ¢ fI &-!&4 0 f -; j + 1 -;9lC-; .&-21 ) m ¢ , = '*>?7R@CB
11. k &-21 ¢ 3hf & a 1 M ¢ 3hf & -21 f;/ F 7 G f9 -21 f;/ ¤
12. k a 1 ¢ 3hf9 & &-21 k -21 0f;/ F 72G f9 ¢ 3hf9 O& -21 0f / ¤
13. k -21 ¢ 3hf9 & a 1 k &-21 ¢ 3hf9 & -21 f / 43V -; £ -;&-21 0f;/ ¤
5
14. k a 1 ¢ 3hf & a 1 k -21 ¢ 3hf & -21 f;/
f9 f / ¤ 15. a 1 ¢ 3hf9 & -21 -21 ¢ 3hf9 & &-21 f /
£ ) 3 m , ¢ 3hf9 & -;&-21 f;/ ¤ 16. k 1 ¢ 3hf9 & &-21 &-21 f;/ 0 ¡ = '8>?7A@CB 17. k &-21 ¢ 3hf9 & 1 ¢ 3hf9 O& &-21 0f / ¡ = '8>?7A@CB 18. k &-21 ¢ 3hf & &-21 f;/ ¡ = '8>?7A@CB 19. k 1 ¢ f9 & &-21 f;/ ¡ = '8>?7A@CB 20. k &-21 ¢ f9 & P f;/ ¡ = '8>?7A@CB 21. k -21 ¢ f9 & Pk &-21 f /
£ F3 7 G -;&-21 0f;/ ¤ 22. k a 1 ¢ f9 & Pk -21 f;/
£ ) 3 m , f / ¤ 23. k &-21 M ¢ f & &-21 f;/
F 7 G f -21 ¢ f & -;&-21 f;/ ¤ 24. k a 1 k -21 ¢ f9 & &-21 0f;/
F 7 G f9 ¢ f9 & -;&-21 f;/ ¤ 1.2. The Hurwitz Zeta Function ζ(ν, z)
1.2.1. Derivatives with respect to the argument
1. fI& 3PfI ; < fI& ¤ 2. EDM &-21 F m GIH f -;&-21 F < m G ¤ 1.2.2. Derivatives with respect to the parameter
1.
& M]! ¤
7 $ £) 3 £ F '*G3 7 $ ' " £! 3 £ 3 ( 7X! " ' ! " &-21 1
' + -21 . F 'PG 3 ( 7! " ] $ (:7X! %O' ! " Y &-21 1
' F £) ' G 7' " 43 £! ¢ = ' ! = @O!C@CB 3.
F 7G - a 1 3 ' ' £ 1A- 3 ¢ 3 £ ¢ = = @ #C7 !C@MB
4.
F 7
G - a 1 C7( " ' ! C7( " ' !0' 3 ']" ' ( 7X! ' ' '#" '#" + &-21 . F 7 G " ' (:7 3 £ ¢ = = @ C7 !C@CB
5.
F 7 G - a 1
C7 (* " ' ! ' 3 C7( " ' ! '%& ' £ ( 7! " '#" '#" + &-21 . F 7 G 3 7( " ' '#" 3 £ ¢ = = @ C7 O!C@CB
6.
F
G - a 1 C7( " ' !AC7<5n " ' ! ' C7 ( " ' ! ' & ' £ C7(* " ' ! '#" ' ( 7X! " "" 5_7X! ']" '#" ']" + &-21 . F 7 G 7 (* " ' 7( " ' 3 £ ¢ = = @& !C@MB
1.3. The Exponential Integral Ei (z)
1.3.1. Derivatives with respect to the argument
1. 3PfI& 3 ¢ &-21 3 ¢ -; - &-21 m ]! "$&% = '8>?7A@ B 2.
3 ¢ -; - -;&-21 fI& = '8>?7A@ B :
!$ "!$ # 3. EDM &-21 F3 m G9H 3Q 3 ¢ -21 -R &-21 m ]! "$&% = '8>?7A@ B 4.
3 ¢ 3 ¢ -21 -R -;&-21 F m G = '8>?7A@ B 5. 3fI&0 f 43fI& f &-21 ( 7X! " $&% m ]! "'& = '8>?7A@ B 6. EDM &-21 R F]3 m GIH 43fI -;&-21 R F]3 m G 43 ¢ f &-21 -; &-21 3 ¢ F m G = '*>h7A@ B
7. -21 - P 3PfI& 3 ¢ -;&-21 43fI& ¤ 8. EDM a 1 -R ED -21 R F3 m G9H H 43 ¢ 4 F]3 m G ¤ 9. V- 3PfI& f 3PfI& ¤ 10. ED -R PED &-21 R F]3 m GIHOH f -;&-21 F]3 m G ¤ 1.4. The Sine si (z) and Cosine ci (z) Integrals
1.4.1. Derivatives with respect to the argument
1. 0fI& ')(:7X! % -; -;&-21 3AfI&3 V- -;&-21 AfI& = '8>?7A@ B
2. EDM &-21 F m GIH 43 ¢ ' (87! % D R -;&-21 F3 m G 3 #- R -;&-21 F m G9H = '*>?7R@CB
3. 0fI& ')(:7X! % -; -;&-21 43 AfI& - -;&-21 AfI& = '8>?7A@ B
4. EDM &-21 F m G9H 43 ¢ ')(87! %O D R -;&-21 F3 m G #- R -;&-21 F m G9H = '*>?7R@CB
5. &<3 0& 3 ¢ FO63 ' G &3 F Q3 ' G &
7 ' 7 ' & 1 3 £) ¢ 3P ¤ !<
; 6. & 0& 43 ¢
F Q3 ' G 0& F Q3 ' G 0&3 7 '%& 1 3 £! 3 ¤
1.5. The Error Functions erf (z) and erfc (z)
1.5.1. Derivatives with respect to the argument
1. 0fI& 43 ¢ &-21 m ' #- W W &-21 0fI& = '8>?7A@ B 2. 0f / ' (87! % m 1 -; - W 1 -;&-21 0f9 & = '8>?7A@ B 3. &-21 0f;/ ( 7! ']" '#" - W &-21 0f;/ = '8>?7A@ B 4. D 7 f;/ H ( 7! ' -;&-21 F 7 f G = '8>?7A@ B 5. DM &-21 F m G9H 3 m ' -;&-21 V- W W &-21 F m G = '8>?7A@ B 6. ED F m GIH 3 "' ']" - W &-21 F m G = '8>?7A@ B 7. D &-21 F m GIH 7 ) 7 m , = '8>?7A@ B 8. EDM &-21 F m G9H 43 ¢ '6(*7X! % m -21 - W 1 -;&-21 ) m , = ' > 7A@ B 9. D W W fI& H 43 AfI W W AfI&3 3 ' & 8 3PfI W W -;&-21 / £ fI ¤ 10. D &-21 W W m H AfI -;&-21 W W F m G
3 3 '%& 8 f -;&-21 W + W . -;&-21 ) m , ¤ 11. D W 0f / H ( m ! ' F 7 G
W F 7 3 _ f9 G ¤
L
! # " ( 12. ED &-21 W F m GIH m ' F 7G -;&-21 W ) 7 3 _ m , ¤ 13. D &-21 V- W D W 0f;/ HOH F 7 G f -21 0f;/ ¤ 14. ED a 1 - W PED &-21 W F m GIHOH
F 7 G f9 -;&-21 F m G ¤ 15. D a 1 W k -21 f / H F 7 G 3Pf9 W 0f / ¤ 16. ED &-21 W ED &-21 F m GIHOH
F 7 G 43f #- W F m G ¤ 17. ED W W 0fI& H 3 ' & 8 3PfI W W -;&-21 / £ fI ¤ 18. ED &-21 W W F m GIH 3 '%& 8 f -;&-21 W + W . -;&-21 ) m , ¤ 19. D &-21 W 0f;/ H "' £ -21 W A - &-21 f / £ ¤ 20. ED -21 W F m GIH
3 ¢ "' £ -;&-21 W + . - &-21 ) f * , ¤ 21. D -21 V- W D &-21 W 0f;/ HOH
43V -; £ -;&-21 0f / ¤ 22. ED a 1 - W ED -21 W F m GIHOH
43V -; £ F m G ¤ 23. ED 1 W &-21 f;/ 0 H ¡ = '8>?7A@ B 24. D a 1 - W D -21 W 0f / HOH f9 f / ¤
P
:1 25. ED &-21 - W ED &-21 W F m G9H H
f -;&-21 F m G ¤ 1.6. The Fresnel Integrals S(z) and C(z)
1.6.1. Derivatives with respect to the argument
1. fI&0 " m ')(:7! % 1 -; D 1 -;&-21 43 AfI&<3 - 1 -;&-21 AfI& H = '*>h7A@ B
2. EDM &-21 gF m GIH 43 ¢ " m ')(:7X! % -!&4 D R 1 -;&-21 F]3 m G 3 #- R 1 -;&-21 F m GIH= '*>?7R@CB
3. 0fI& " m ')(:7! % 1 -; D 1 -;&-21 43 AfI& #- 1 -;&-21 AfI& H = '*>h7A@ B
4. DM &-21 F m G9H 43 ¢ " m ')(:7X! % -!&4 D R 1 -;&-21 F]3 m G - R 1 -;&-21 F m GIH= '*>?7R@CB
1.7. The Generalized Fresnel Integrals S(z, ν) and C(z, ν)
1.7.1. Derivatives with respect to the argument
1. fI ;0 3 ')(:7X! % f# -;
-;&-21 3AfI&3 - -;&-21 AfI& = '*>?7R@CB
2. D &-21 F m GIH 43 ¢ &-21 ')(:7! % fV - -21 D R -;&-21 F3 m G 3 - R -;&-21 F m G9H= '*>?7R@CB
3. 0fI 3 ')(:7X! % f# -;
-;&-21 3AfI& - -;&-21 AfI& = '*>?7R@CB
,9N
4. EDM &-21 F m G9H
43 ¢ &-21 ')(:7! % fV - -21 D R -;&-21 F 3 m G 3 - R -;&-21 F m G9H= '*>?7R@CB 1.8. The Incomplete Gamma Functions γ(ν, z)
and Γ(ν, z)
1.8.1. Derivatives with respect to the argument
1. fI&0 3 ¢ fV -; - -;&-21 fI& = '8>?7A@ B 2. EDM &-21 F m GIH 3 ¢ 3 ¢ f# - -21 -R -;&-21 F m G = '8>?7A@ B 3. - fI&0 43 ¢ - -; < fI& ¤ 4. EDM a -21 F m G9H -21 F < m G ¤ 5. fI&0 ¢ 3 3PfI )3 < fI& ¤ 6. DM &-21 R F m G9H ¢ 3 f -;&-21 R F T3 _ m G ¤ 7. &- fI&0 ' %
f 1 N 1 J '65_7 ! m 5_7 L ¤ 8. D -21 R F m G9H 43 ¢ ' %
,
L1 , 18. ED &-21 F m G9H 43 ¢ &-21 3 ¢ fV - -21 -R -;&-21 0fI& = '8>?7A@ B 19. - fI& 3 ¢ - -; - _ fI& ¤ 20. ED a -21 F m GIH -21 F < m G ¤ 21. fI&0 ¢ 3 ; 43fI T3 _ fI& ¤ 22. D &-21 R F m GIH ¢ 3 ; f -;&-21 R F T3 < m G ¤ 23. &- fI& ¢ 3 ; f J '65_7 ! m 5_7 L ¤ 24. D -21 R F m GIH 3 ¢ ¢ 3 ; f -;&-21 ^ '65_7 ! 5_7 ` ¤ 25. ¢ 3 _ fI& 3 ¢ " m 1 -; - A &-21 F m G ¤ 26. ED &-21 F ¢ 3 < m GIH " m -!&4 -R + . &-21 F mOG ¤ 1.8.2. Derivatives with respect to the parameter
1.
1.9.1. Derivatives with respect to the argument
1. fI&0 F]3 m G J ' $ L £ 43 ; &- F m :G - 0fI& ¤ 2.
, ,
8. EDM a + -21 . - W A 0f;/ H 43 £ -; I+ -21 . - W A a 0f;/ ¤ 9. D W + . F m G9H
3 £ -; 43 -; a W + . - F m G ¤ 10. ED - + a 1 . - W + . F m G9H
£ -; -;&- + a 1 . - W + . a F m G ¤ 11. D -21 V- W A - f;/ H 3 £ -; -;&-21 V- W A ¤ 12. ED &-21 - W + . - F m G9H £ -; -21 - W + . ¤ 13. ED -21 - W A - &-21 f;/ H 3 ¢ £ -;&-21 O/ -;&-21 F f " G ¤ 14. ED - W + . - &-21 F m G9H £ -;&-21 O/ F m G ¤ 15. ED W A 0f / H
43 ¢ W A &-21 '65 $ ! ')( $ ! $9% ! '%&" f &- 3 ; &- -; a 0f;/ = '*>?7R@CB
16. ED V- W A f;/ H 43 ¢ - W A &-21 '65 $ !
')( $ ! $&% ! '%&" f &- a &- f;/ = '*>?7R@CB 17. D a 1 V- W A D -21 W A 0f;/ H H
F 7 ( G ) m , - W A 0f / ¤ 18. D &-21 - W + . P D &-21 W + . F m GIHOH
F 7 ( G ) m , -;&-21 - W + . F m G ¤ 19. D &-21 - W A D W A 0f / HOH
F]3 G ) m , -21 - W A f;/ ¤ ,93
P1 , 20. ED a 1 - W + . PED &-21 W + . F m GIHOH
F]3 G ) m , -;&-21 - W + . F m G ¤ 21. D &-21 W A D - W A 0f / HOH
F 5_7 G ) 3 m , -21 W A f;/ ¤ 22. D a 1 W + . D &-21 V- W + . F m GIHOH
F 5_7 G ) 3 m , -;&-21 W + . F m G ¤ 23. ED a 1 W A ED -21 #- W A 0f;/ H H
F ¢ G ) 3 m , W A 0f / ¤
24. ED &-21 W + . ED &-21 - W + . F m GIHOH F ¢ G ) 3 m , -;&-21 W + . F m G ¤
25. ED a 1 - W A ED &- -21 W A f;/ H H 3# -; 3 I+ -21 . -; - W A f;/ ¤
26. D &- -21 - W + . D W + . F m GIHOH 43V -; 3 ; - + a 1 . - W + . F m G ¤
27. D - -21 W A D a + -21 . - W A f / HOH 43# -; ¢ -;&- -21 W A f / ¤
28. ED a a 1 W + . EDM - + a 1 . V- W + . F m GIHOH 43# -; ¢ W + . F m G ¤
1.9.2. Derivatives with respect to the order
1.
M]! £ -;&-21 -
W F G Y 3 C 3 £ F 7 3 G F G 3h N
7 #7 ! W & #$
3 43 ¢ £ &-21 - W 1
"
#$
"
M]! a 1 £ -;&-!&4 - W a 1 F G Y
#$
3 ¢ £ -;&- £ ¢ -21 - W 1 ( " $ (:7! % 'T( $ 5_7X! % F 7 3 G
3 ¢ £ &-!&4 - W 1 7$
-21 &- ) , £ ¢ -!&4 &- a 1 ) , Y "" "" J 1 L)" 1 N 1
$ ! W $ 5 1 #$ 3 /
W 1 - -21 ) 3 , 3 43 ¢ £ &-21 / W -;&-21 ) 3 ,
3 ¢ ' % J & L ' a 1#-
W 1 N 1 '65_7 ! W 'g5 &
#$ 3 ' & V- W F]3 3 7 G ¤ 3.
& & -1 & N & 7 #7 ! 1
&
#$
£ &4 C £ £ -21 ) , 1 ) , 3
£ C £ 1 ) , 83 £ C 3 £ -21 ) , = @CB 1.10. The Bessel Function Jν(z)
1.10.1. Derivatives with respect to the argument
1. fI&0 F m G 3 ¢ J ' $ L 2 0fI& ¤ , ;
+ ( N1 2.
43& -; ( m ]! "')( $ ! % &- m ! "
% $ ( ! % a - 0fI& ¤ 3. EDM &-21 F m GIH F m G -;&-21 43 ¢ J ' $L 2 F m G ¤ 4. f;/ 0 F]3 7 G ¢ J ' $ L F G &- F m G f;/ ¤ 5. f / 0 F m G I+ -; . 2 f / ¤ 6. I+ a 1 . a 1 0f / 0 7 F m G &-21 f / ¤ 7. I+ a 1 . -;&-21 0f / ( 7! ' F m G &-21 0f / ¤ 8. - + a & . a 1 0f;/
/ *F m G a 1 a 1 F m G -;&-21 F m G ¤ 9. f / 0
F m G + -!&A . &-21 fI - '65 $ !$&% ')( $ ! - 2 0f / = '*>?7R@CB 10. ED &-21 F m GIH 7 ¢ J ' $ LF G &- F m G F m G ¤ 11. ED -21 F m GIH F m G - + . -21 F m G ¤ 12. EDI+ &- . a 1 F m GIH ( 7X! ' F m G &-21 -;&-21 F m G ¤ 13. EDI+ &- . -;&-21 F m GIH 7 F m G &-21 -;&-21 F m G ¤ 14. EDI+ &-21 . a 1 F m GIH
43 ¢ / 8F m G a 1 -21 a 1 F m G -;&-21 F m G ¤ 15. &-21 &-21 fI&0 m ! '#" &-21 = '8>?7A@ B
,9:
16. D -21 R &-21 F m G9H ( 7! ' £ fI &-21 - R = '8>?7A@ B 17. &-21 fI& &-21 fI&0 m ! '#" &-21 £ fI& ¤ 18. &-21 0fI& &-21 fI&0 m ! ']" &-21 £ fI& = '8>?7A@ B 19. &-21 fI& 1 -; fI&0 3 ¢ a 1 m ! '#" &-21 £ fI& = '*>?7A@ B 20. &-21 0fI& 1 -; fI&0 3 ¢ m ! '#" &-21 £ fI& ¤ 21. D -21 m &-21 F m GIH ( 7X! ' £ fI &-21 - m ¤ 22. ED -21 m &-21 F m GIH ( 7X! ' £ fI &-21 - m = '8>?7A@ B 23. 0f;/ F m G 43 ¢ J ' $ L a f;/ -; a 0f;/ ¤ 24. &-21 &-21 0f;/ m ']" I+ &-!& . &-21 £ f;/ = '8>?7A@ B 25. &-21 1 -; 0f / 3 m ']" I+ &-!& . &-21 £ f / = '8>?7A@ B 26. &-21 &-21 0f;/ 1 -; 0f;/
m '#" I+ &-!& . 1 -; £ f;/ = '*>?7R@CB 27. &-21 0f / a 1 0f / m '#" I+ &-21 . a 1 £ f / ¤ 28. D -21 &-21 F m GIH ( 7X! ' f &-21 - + a 1 . &-21 F m G = '*>?7R@CB 29. D -21 1 -; F m GIH ( 7X! '%& f &-21 - + a 1 . &-21 F m G= '*>?7R@CB 30. D -21 &-21 F m G 1 -; F m G9H
( 7X! ' f &-21 - + a 1 . 1 -; F m G ¤ , <
+ ( N1 , 31. ED -21 &-21 F m G a 1 F m G9H
( 7! ' f &-21 - + a & . a 1 F m G ¤ 32. ED &-21 F m GIH
F3 m G -; -21 43 ¢ J ' $L a F m G -; a F m G ¤ 1.10.2. Derivatives with respect to the order
1.
2.
M]! -21 *
£ & £ &0 = = @ B VC7 !C@MB 4.
M]! a 1 £ & a 1 0&3 43 ¢ £ & -;&-21 & ' % F G &-21 J L "$&% ' ( $ ! a 1 0&3 ' % 1 J L
"')( $ ! % $ Y -21
% &- a 1 & -21 £ &<3 3 ¢ &- - -;&-21 0& 1 - £ &0 ¤ 5.
M]! 1 -; £ & 1 -; &<3 3 ¢ £ & &-21 &
3 ' % &-21 J ( L ""'$&% ')( $ ! 1 - &<3 ' % 1 "')( $ ! % $
Y -21 ")"'&
% 3 ¢ -; a 1 0& -21 £ &3 3 ¢ a &- -21 & 1 - £ &0 ¤ 6.
M]! 1 ¢ &-21 £ a 1 / -;&-21
Y &-21 ')( $ ! % J L "
$9% ')(* $ ! % J $ 5 1 L J $ (*'65 1 L DXF F 7G 3 F 3 7GG ,9L
Y
£ & H ¢ £ &-21 / -; a 1 &-21 ')( $ (:7X! % J L "
$&% ')(* $ (:7! % J $ 5 & L J $ (*'65 1 LY DXF F G 3 F 3 7GG 3 £ &
£ & H = '*>h7A@CB 1.11. The Bessel Function Yν(z)
1.11.1. Derivatives with respect to the argument
1. 0fI& F m G 3 ¢ J ' $ L 2 0fI& ¤ 2.
43& -; ( m ]! "')( $ ! % &- m ! "
% $ ( ! % a - 0fI& ¤ 3. D &-21 F m GIH F m G -;&-21 43 ¢ J ' $ L 2 F m G ¤ 4. f / 0 F m G I+ -; . 2 0f / ¤ 5. I+ a 1 . a 1 0f;/ 3 7 F m G &-21 f;/ ¤ 6. I+ a 1 . -;&-21 0f / ( 7X! ' F m G &-21 f / ¤ 7. - + a & . -;&-21 0f /
43 ¢ a 1 / 8F mOG a 1 a 1 F m G -;&-21 F m G ¤ 8. f / 0 F m G I+ -!&A . &-21 fI - '65 $ !$9% ')( $ ! - 2 f / = '*>h7A@CB
9. D -21 F m G9H F m G I+ -; . -21 F m G ¤ 10. EDI+ &- . a 1 F m GIH ( 7X! '%& F m G &-21 -;&-21 F m G ¤
,9P
+ ( 11. EDI+ &- . -;&-21 F m G9H 7 F m G &-21 -;&-21 F m G ¤ 12. D + &-21 . -;&-21 F m G9H
3 / *F mG a 1 -21 a 1 F m G -;&-21 F m G ¤ 13. &-21 1 -; fI&0 3 ¢ a 1 m ! '#" &-21 = '8>?7A@ B 14. D -21 R 1 -; F m GIH 3 m ! '#" - R = '8>?7A@ B 15. &-21 1 -; 0f;/ 3 ¢ a 1 m ']" I+ &-!& . 1 -; £ f;/ = '*>?7R@CB 16. &-21 &-21 0f;/ 1 -; 0f;/
3 ¢ a 1 m ']" I+ &-!& . &-21 £ f;/ = '8>?7A@ B 17. &-21 0f;/ a 1 0f;/
43 ¢ a 1 m '#" I+ &-21 . -;&-21 £ f;/ ¤ 18. ED -21 1 -; F m GIH 3 m '#" - + a 1 . 1 -; F m G = '8>?7A@ B 19. D -21 &-21 F m G 1 -; F m G9H
3 m ']" - + a 1 . &-21 F m G ¤ 20. ED -21 &-21 F m G a 1 F m G9H
3 7 f &-21 - + a & . -;&-21 F m G ¤ 21. &-21 &-21 0f / 1 -; 0f /
3 ¢ a 1 m ']" I+ &-!& . &-21 £ f / = '8>?7A@ B 22. &-21 1 -; 0f;/ &-21 0f;/
3 ¢ a 1 m ']" I+ &-!& . &-21 £ f;/ = '8>?7A@ B 3 N
, 3 8 , 3 8
23. &-21 &-21 0f;/ &-21 0f;/ ( 7X! ' f &-21 I+ &-!& . 1 -; £ f / ¤
24. &-21 0f / -;&-21 0f / ( 7X! ' f &-21 I+ &-21 . a 1 £ f;/ ¤
25. a 1 0f;/ 1 -; 0f;/ ( 7X! '%& f &-21 I+ &-21 . a 1 £ f;/ ¤
26. D -21 &-21 F m G &-21 F m G9H m ']" - + a 1 . 1 -; F m G ¤
27. ED -21 1 -; F m G &-21 F m G9H 3 m ']" - + a 1 . &-21 F m G ¤
28. D -21 &-21 F m G -;&-21 F m GIH m '#" - + a & . a 1 F m G ¤ 29. ED -21 a 1 F m G 1 -; F m G9H
3 m ']" f &-21 - + a & . a 1 F m G ¤ 1.11.2. Derivatives with respect to the order
1.
2.
ν (z) and H(2) ν (z)
1.12.1. Derivatives with respect to the argument
1. + . f / 0 F m G I+ -; . + . 2 0f / = 7]X@CB
2. EDM -21 + . F m G9H F m G I+ -; . -21 + . F m G = 7]X@CB
31
+ ( !, , 3. I+ a 1 . + . a 1 0f / ( 7X! F m G &-21 + -21 . U = 7]X@CB 4. DM &-21 + 1 .1 -; 0f;/ + .1 -; f;/ H ¡ = '8>?7A@CB 1.12.2. Derivatives with respect to the order
1.
3 8 M]! 3 ¢ + . & ' % &-21 J L ""'$&% ')( $ ! + . 0& = 7]X@CB
2.
3 8 M]! -; 3 ¢ a + . &<3 3 ¢ ' % &-21 J L
""'$9% ')( $ ! + . &= 7@CB 3.
3 8 M]! 1 *
Y + -21 . 3 ¢ a 1 £ &<3 £ & 43 ¢ = 7@CB 4.
3 8 M]! -21 *
Y + -21 . £ & 3 ¢ a 1 £ &0 3 ¢ = 7@CB 1.13. The Modified Bessel Function Iν(z)
1.13.1. Derivatives with respect to the argument
1. fI&0 F m G J ' $L 2 0fI& ¤ 2.
43& -; ( m ]! "')( $ ! % &- m ]! " % $ ( ! % - a 0fI& ¤
3. DM &-21 F m GIH F3 m G -;&-21 J ' $ L 2 F m G ¤ 4. f;/ 0 F]3 7G J ' $L F G &- F]3 m G f;/ ¤ 5. f;/ 0 F m G I+ -; . 2 0f;/ ¤ 6. I+ a 1 . a 1 f / 0 7 F m G &-21 0f / ¤
3,
7. I+ a 1 . -;&-21 f / 0 7 F m G &-21 0f / ¤ 8. - + a & . a 1 f;/ 0
/ 8F mO G a 1 a 1 F m G -;&-21 F m G ¤ 9. k 0f /
F m G I+ -!&A . &-21 3PfI - '65 $ !$&% ' ( $ ! - 2 f / = '*>?7R@CB 10. D &-21 F m G9H 7 J ' $ L F G &- F3 m G F m G ¤ 11. ED -21 F m G9H F]3 m G - + . -21 F m G ¤ 12. EDI+ &- . a 1 F m GIH ( 7! ' F m G &-21 -;&-21 F m G ¤ 13. EDI+ &- . -;&-21 F m GIH ( 7X! ' F m G &-21 -;&-21 F m G ¤ 14. D I+ &-21 . a 1 F m G9H
3 ¢ / *F m G a 1 -21 a 1 F m G -;&-21 F m G ¤ 15. &-21 &-21 0fI& m ! '#" &-21 = '8>?7A@CB 16. k &-21 - a 1 fI& m ! "'#" -;&-21 £ ¢!!£ fI& ¤ 17. k &-21 fI& £ fI # -21 J 5n'65 1 L 5_7X! 1 N 1 5n'65 1 5_7 ! m ¤ 18. ED -21 R &-21 F m GIH ( 7X! ' £ fI &-21 - R = '8>?7A@CB 19. D -21 V-R a 1 m H 43 ¢ m ! "'#" F £ ¢) m G ¤ 20. ED 7 R F m GIH
#$ ¤
3 3
+ ( 31 21. &-21 0fI& &-21 0fI& m ! ']" &-21 £ fI& ¤ 22. &-21 0fI& &-21 0fI& m ! '#" &-21 £ fI& = '8>?7A@CB 23. ED -21 m &-21 F m GIH ( 7! ' £ fI &-21 - m ¤ 24. D -21 m &-21 F m G9H ( 7! ' £ fI &-21 - m = '8>?7A@CB 25. V f / F m G J ' $ L a 0f / -; a 0f / ¤ 26. &-21 V &-21 f;/ 0 m '#" I+ &-!& . &-21 £ f;/ = '8>?7A@CB 27. &-21 V 1 -; f / 0 m '#" I+ &-!& . &-21 £ f / = '8>?7A@CB 28. &-21 &-21 f;/ 1 -; f;/ 0
m ']" I+ &-!& . 1 -; £ f;/ = '*>?7R@CB 29. &-21 0f / a 1 f / 0 m ']" I+ &-21 . a 1 £ f / ¤ 30. ED &-21 F m GIH
F3 m G -; -21 KJ ' $L a F m G -; a F m G ¤ 31. D -21 V &-21 F m G9H ( 7! ' f &-21 - + a 1 . &-21 F m G = '*>?7R@CB 32. ED -21 1 -; F m G9H ( 7! ' f &-21 - + a 1 . &-21 F m G = '*>?7R@CB 33. ED -21 &-21 F m G 1 -; F m GIH
( 7X! ' f &-21 - + a 1 . 1 -; F m G = '8>?7A@CB 34. ED -21 &-21 F m G a 1 F m GIH
( 7! ' f &-21 - + a & . a 1 F m G ¤ 395
35. a - P - 0fI& F 7G £ fI - 0fI& ¤ 36. ED &- - R PED a &-21 R F m GIHOH
F 7 G £ fI - -;&-21 -R F m G ¤ 37. &- P - 0fI& F 7 3 G 3 £ fI - fI& ¤ 38. ED a R ED &- -21 V-R F m GIHOH
F 7 3 G 3 £ fI -;&-21 R F m G ¤ 39. &- - < 0fI& F 7 3 G £ fI - - 0fI& ¤ 40. D a #- R D &- -21 R F m GIHOH
F 7 3 G £ fI -;&-21 -R F m G ¤ 41. a P - - 0fI& F 7 G 3 £ fI fI& ¤ 42. D &- R P D a -21 -R F m GIHOH
F - 7G 43 £ fI - -;&-21 R F m G ¤ 1.13.2. Derivatives with respect to the order
1.
M]! 1 * 7 3 £ & #- £ & = = @ B C7 !C@MB
3. *
M]! a 1
7 3 £ & -;&-21 & a 1 &03 ( 7X! ' £ & a 1 0& 643 ¢ ' % &-21 ( 7X! "$9% ')( $ ! F ,G -; a 1 &
3 ' % " 1 J (
L "')( $ ! % $ -21
( ! %Y 3 ¢ &- a 1 & -;&-21 0& -21 £ &3 43 ¢ &- &- a 1 & -21 £ & 1 - £ &0 ¤3;
+ ( 5) 5.
7 £ &3 £ &00 -;&-21 & a 1 &0&3 ( 7! ' a 1 &Y £ & £ & 43 ¢ ' % &-21 ( 7X! "$&% ')( $ ! F G -; a 1 0&
3 ' % " 1 J (
L "')( $ ! % $ -21
(]! %Y 3 ¢ &- a 1 & -;&-21 0& -21 £ &3 43 ¢ &- &- a 1 & -21 £ & 1 - £ &0 ¤
6.
M]! 1 -;
7 £ &<3 £ &00 -;&-21 & a 1 &03 ( 7! ' £ & £ & &-21 &
3 ' % &-21 J ( L ""'$9% ')( $ ! 1 - & ' % " 1 J L
"')( $ ! % $ -21
%Y 3 ¢ -21 &- -21 & -; a 1 0& -21 £ & 43 ¢ &- &- -21 & -21 £ & 1 - £ &0 ¤ 7.
M]! 1 £ &-21 O/ -;&-21 &-21 ( 7! " ')( $ ! % J L
"
$9% ')(* $ ! % J $ 5 1 L J $ (*'65 1 LY D £ F F 7 G 3 F 3 7 G;G 3 £ & V- £ & H
£ &-!&4 / -; a 1 &-21 ( 7! " ')( $ (:7! % J L "
$9% ')(* $ (:7X! % J $ 5 & L J $ (*'65 1 LY D £ F F G 3 F 3 7GG 3 43 £ & - £ & H ¤
1.14. The Macdonald Function Kν(z)
1.14.1. Derivatives with respect to the argument
1. fI&0 F]3 m G J ' $ L 2 0fI& ¤ 2.
43& -; m ]! "')( $ ! % &- ( m ! " % $ ( ! % - a 0fI& ¤
3 :
$ 3. EDM &-21 F m GIH F m G -;&-21 J ' $ L 2 F m G ¤ 4. f;/ F3 m G I+ -; . 2 0f;/ ¤ 5. I+ a 1 . a 1 f;/ 0 43 ¢ '%& f &-21 - ¤ 6. - + a & . a 1 f;/ 0 ( 7! ' F m G a 1 -;&-21 a 1 F m G ¤ 7. f /
F3 m G I+ -!&A . &-21 f - '65 $ !$&% ')( $ ! - 2 f / = '*>?7R@CB 8. EDM -21 F m GIH F m G - + . -21 F m G ¤ 9. DM + &-21 . a 1 F m G9H 7 F m G a 1 -21 a 1 F m G ¤ 10. k &-21 a 1 0fI& 3 ¢ £ / 6 £ fI -;&-21 -;&-21 ¤ 11. k SEa 1 )SEa 1 fI& 3 ¢ SEa _ / g £ fI &- S -21 &- S -21S -; £ fI& = > 'I@ B 12. k &-21 )SEa 1 fI& 43 ¢ S / 6 £ fI - S -21 - S -21 - S -21S -; £ fI& = > 'I@ B 13. k - S -21 )SEa 1 fI& 3 ¢ SEa / g £ fI - S -21 - S -;&-21 - S -;&-21S £ fI& ¤ 14. k SEa 1 - )SEa 1 fI& 3 ¢ SEa _ / g £ fI &- S -21 - &- S -21S £ fI& ¤ 15. k &-21 - )SEa 1 fI& 43 ¢ S / 6 £ fI - S -21 - S -21 - - S -21SEa £ fI& ¤ 16. k - S -21 #- )SEa 1 fI& 3 ¢ S / g £ fI - S -21 - S -;&-21 #- - S -;&-21SEa £ fI& ¤ 17. D -21 R )SEa 1 F m GIH
43 ¢ SEa / 6 £ fI - S -21 S -; - S -21S -; F m G = >_'I@CB 3<
+ ( 5) 18. ED SEa &-21 R )SEa 1 F m GIH
3 ¢ S / g £ fI - S -21 S - S -;&-21S F m8G ¤ 19. ED &- S -!&4 R )SEa 1 F m G9H
3 ¢ S ? / g £ fI &- S -21 -;&-21 &- S -21S -; F m G = > 'I@ B 20. D -21 V-R )SEa 1 F m GIH
3 ¢ SEa / g £ fI - S -21 S -; - R - S -21SEa F m G ¤ 21. ED SEa &-21 -R )SEa 1 F m G9H
3 ¢ SEa / g £ fI - S -21 S - R - S -;&-21SEa F m:G ¤ 22. ED &- S -!&4 -R )SEa 1 F m G9H
43 ¢ S _ / g £ fI &- S -21 -;&-21 V- R &- S -21S F m G ¤ 23. 0f;/ 0 F]3 m G J ' $ L a f;/ -; a 0f;/ ¤ 24. &-21 &-21 f;/ 0 43 ¢ / f &-21 + &-!& . &-21 £ f;/ ¤ 25. &-21 0f;/ a 1 f;/ 0
43 ¢ / f &-21 I+ &-21 . a 1 £ f;/ ¤ 26. ED &-21 F m G9H
F m G -; -21 J ' $ L a F m G -; a F m G ¤ 27. D -21 &-21 F m G9H / f &-21 - + a 1 . &-21 F m G ¤ 28. D -21 &-21 F m G a 1 F m GIH
/ f &-21 O - + a & . a 1 F m G ¤ 29. &-21 &-21 f / &-21 f / 0
m ']" I+ &-!& . &-21 £ f;/ = '*>?7R@CB 3 L
$ 30. ED -21 &-21 F m G &-21 F m GIH
( 7X! ' f &-21 - + a 1 . &-21 F m G = '8>?7A@CB 31. - &-21 fI&0 3 ¢ F 7 3 G F 7 G -;&-21 - 0fI& ¤ 32. ED - R PED -21 R F m GIHOH
3 ¢ F 7 3 G F 7 G -21 V-R F m G ¤
33. < &-21 - fI&0 43 ¢ F 7 3 G F 7 G -;&-21 0fI& ¤ 34. ED R PED -21 -R F m GIHOH
3 ¢ F 7 3 G F 7 G -21 R F m G ¤
35. &- S -21 - SEa 1 )SEa 1 fI&0 3 £ fI - S -21 #- )SEa 1 fI& ¤ 36. ED SEa a 1 - R PED &- S -!&4 R )SEa 1 F m G9HOH 43 £ fI S -;&-21 -R )SEa 1 F m G ¤ 37. a SEa 1 < - S -21 - )SEa 1 fI&0 5n'! % % 43 £ fI SEa 1 )SEa 1 fI& ¤ 38. ED &- S -21 R PED a2S -21 -R )SEa 1 F m G9HOH
5n'! % % 3 £ fI - S -;&-!&4 R )SEa 1 F m G ¤ 39. a SEa 1 - P - S -21 )SEa 1 fI&0 5h'! % % £ fI SEa 1 - )SEa 1 fI& ¤ 40. ED &- S -21 - R PED a2S -21 R )SEa 1 F m G9HOH
5n'! % % £ fI - S -;&-!&4 V-R )SEa 1 F m G ¤ 41. - &-21 )SEa 1 fI&0 5n'! % (*'! % -;&-21 - )SEa 1 fI& ¤
3 P
+ ( 5) , 42. ED - R PED -21 R )SEa 1 F m GIHOH
5n'! % (*'! % -21 -R )SEa 1 F m G ¤ 1.14.2. Derivatives with respect to the order
1.
M! ¡\¤
2.
M!
4.
' % &-21 J L ""'$&% ' ( $ ! -21 0&3 43 ¢ ' % / 1 J ( L
"')( $ ! % $ &- -21 0& -; a 1 0& -21
!& % -21 £ & ¤
6. 3 ¢ £ &<3 £ &00 &-21 & 1 -; 0& ' % &-21 J L ""'$9% ')( $ ! -21 &3 3 ¢ ' % / 1 J (
L "')( $ ! % $ &- -21 &
% -21 £ & ¤ 7.
"
"
$9% ')(* $ (:7X! % J $ 5 & L J $ (*'65 1 LY D - F F G 3 F 3 7 GG 3 3 £ & H ¤ 5N
H
L
1. H fI&0 F3 G J ( 1 L
"$&% ')(* $ ! % &- J ')( $
L<F G &- - F]3 m nG H - 0fI& ¤ 2.
F]3 G J ( 1 L "$9% ')(* $ ! % &-
J ' ( $ L F]3 G &- - Y F m nG H a 0fI&3 7
F m nG a -21 -21
J 5 1 L
¤
3. H f;/ 0 F m G I+ -; . H -; f;/ ¤ 4. - H f / 0 F3 m G - + a . H a 0f /
3 ( 7X! ' F m G a &-21 -21 &-21 J $ 5 1 L J 5n')( $ 5 1 L F
m G ¤ 5. + a 1 . H a 1 f;/ 0 F m G &-21 m ¤
6. EDM &- -21 H F m G9H F]3 m G - + a . -21 H -; F m G ¤ 7. D I+ &- . H a 1 F m G9H 43 ¢ F m G &-21 -;&-21 m ¤ 8. DM a -21 H F m G9H F m G + -; . -21 H a F m G
3 7 F m G a &-21 -;&-21 &-21 J $ 5 1 L
J 5:' ( $ 5 1 L F m G ¤ 9. L 0fI&
F]3 G J ( 1 L "$&% ')(* $ ! % &-
J ')( $ L F G &- - F]3 m nG L - 0fI& ¤
5)
+ ( !; , 10.
F 3 G J ( 1 L "$&% ')(* $ ! % &-
43 ¢ J ')( $ L F 3 G &- - Y F m G L a fI& 7
F m G a -21 -21
J 5 1 L J 5 Q(
5 1 L F]3
m G
¤
11. L 0f / F m G I+ -; . L -; f / ¤ 12. - L 0f;/ F m G - + a . L a f;/
7 F m G a &-21 -1 &-21 J $ 5 1 L
J 5n')( $ 5 1 L F 3 m G ¤
13. I+ a 1 . L a 1 0f;/ F m G &-21 m ¤
14. ED &- -21 L F m GIH F3 m G - + a . -21 L -; F m G ¤ 15. EDI+ &- . L a 1 F m GIH 3 ¢ F m G &-21 -I&-21 m ¤ 16. ED a -21 L F m GIH F3 m G I+ -; . -21 L a F m G
( 7X! ' F m G a &-21 -;&-21 &-21 J $ 5 1 L
J 5n')( $ 5 1 L F3 m G ¤ 1.15.2. Derivatives with respect to the order
1.
H M! 3 & ']" "'
&
<!¡
7 &-21 J 1 L)"J 1 L ']")" F G &- -21 D 3 F 3 7 GIH
' % &-21 3 ¢ J L ""'$&% ')( $ ! H - & = Q> @CB 5,
H
L
3.
&
!_!¡
H M! 1 *
Y C £ &3 £ &0 £ &3 £ 0& = = @ B #O !C@CB 5.
H M! -1 *
Y £ &3 £ &0&3 £ &<3 £ &0 = = @ B # !C@CB 6.
H M]! a 1 £ &3 £ &0 a 1 0& 3 ¢ £ &3 £ 0& -I&-21 0& H a 1 &<3 a 1 &0 7 F G a 1 F 7G D C £ £ F 7 3 GIH
3 ' % F G &-21 J ( L "$&% ' ( $ ! - -21 0&3 ' % F G a 1 &-21 J 1 L)"$9% ')( $ ! 3 J L ']"
&-21 J 1 L"')( $ ! % F G 3 ¢ 3 / 8F G 1 -; &-21 J L "$&% ' ( $ ! &- -21
J L %Y 43 ¢ a 1 a 1 0& 1 - &<3 £ -21 1 - £ & 3?43 ¢ - -21 & -21 0&3 £ -21 -21 £ & ¤ 7.
L M! &<3 7
¡
3 7 &-21 43 ¢ J 1 L "J 1 L'#")" F G &- -21 D 3 F 3 7 G9H
' % &-21 J ( L ""'$9% ')( $ ! L - 0& = Q> @CB 12.
L M]! -; 3 ¢ & ( ! '#" "'
L M]! 1 *
3 7 7 F C ,G - £ &<3 £ 0& 3 43 £ &<3 £ 3P& = = @ B C7 !C@MB 14.
L M]! 1 *
15.
£ &<3 £ &03 £ &<3 £ &0 ¤ 16.
L M]! a 1 £ &3 £ &0 a 1 0&3 £ &<3 £ &0 -;&-21 0& L a 1 &<3 -;&-21 0& ( 7X! '%& F G a 1 F 7 G D C £ £ F 7 3 GIH
' % F3 G &-21 J ( L "$9% ')( $ ! - -21 & 43 ¢ ' % F G a 1 &-21 J 1 L "$9% ')( $ !
5 5
J L '#" &-21 3 ¢ J 1 L "')( $ ! % F G 3 ¢
43 ¢ / 8F ,G 1 -; &-21 J ( L "$&% ' ( $ ! &- -21 J ( L %Y a 1 & 1 - &<3 £ -21 1 - £ & 3 - -21 0& -21 &<3 £ -21 -21 £ & ¤
17.
L M]! -I&-21 £ &<3 £ & a 1 0& £ &<3 £ &0 -;&-21 & 3 ' % &-21 J ( L
""'$9% ')( $ ! a 1 & ' % / 1 J ( L "')( $ ! % $ -21
( ! %Y &- a 1 & £ 1 - -21 &<3 -21 £ & 3 -;&-21 0& £ 1 - 1 - &<3 1 - £ & ¤
1.16. The Anger Jν(z) and Weber Eν(z) Functions
1.16.1. Derivatives with respect to the argument
1. J 0& F3 ,G -; J ( 1 L "$9% ')(* $ ! % &-
J ')( $ L F G \ \ F]3 ,G
Y J - &<3 3 ¢ ! -21
F Q( ( 5 7 G
F G ¤
"$&% ')(* $ ! % &- J ')( $
5 5_7 G
F]3 G ¤
3. F 7G 43 ¢ J ' $,L J 2 & ¤
4. J 0f;/ F m G + -; . J -; 0f;/ 3 F3 m G ! m I+ -;&-21 . &-21 F ' ( $ ( 5 7 G F m G ¤
5;
+ ( :1 5. - J 0f;/ F3 m G - + a . J a f;/
3 F m G ! m - + a a 1 . &-21 F ')( $ 5 5_7 G F]3 m G ¤ 6. EDM &- -21 J F m GIH F3 m G - + a . -21 J -; F m G
3 F m G ! m - + a a 1 . &-21 F ')( $ ( 5_7 G F m G ¤ 7. D a -21 J F m GIH F m G I+ -; . -21 J a F m G
3 F]3 m G ! m I+ -;&-21 . &-21 F ')( $ 5 5_7 G F]3 m G ¤ 8. E &0 F 3 G -; J ( 1 L
"$9% ')( $ ! % &- J ')( $
E - & 7 -21
F G ¤
"$&% ')(* $ ! % &- J ')( $
E a & 7 -21
F G ¤
10. F 7G 43 ¢ J ' $ L E 2 & ¤
11. E f;/ 0 F m G + -; . E -; 0f;/ 7 m F m G + -;&-21 . Y &-21 3 ¢ 43 ¢ XF ')( $ ( 5_7 G F m G ¤
12. - E f;/ 0 F]3 m G - + a . E a 0f;/ 7
m F m G - + a a 1 . Y &-21 3 ¢ 3 ¢ XF ')( $ 5 5_7 G F m G ¤ 5:
Jν(z)
Eν(z)
13. D &- -21 E F m G9H F]3 m G - + a . -21 E -; F m G F3 m G " 3 '&(' & 8
mY &-21 3 ¢ 3 ¢ XF ')( $ ( 5_7 G F m G ¤ 14. ED a -21 E F m G9H
F m G I+ -; . -21 E a F m G F 3 m G 3 "']" 8 mY &-21 3 ¢ 43 ¢ XF ')( $ 5 5_7 G F m G ¤
1.16.2. Derivatives with respect to the order
1.
' % &-21 J L '#")"$9% ')( $ ! 0& H &
3 7 &-21 J 1 L "J 1 L '#")" F ,G &- -21 ( 7X! ' &-21 3 ¢ F ')( $ 5_7 G F G ¤ 2.
J M! -I 43 ¢ &-21 ' % &-21 J L
']")"$&% ')( $ ! & H -; & ( 7X! ' &-21 F ')( $ 5_7 G F G ¤
3.
-21 J 1 L J 1 L)""
F ,G - -21 7 &-21 3 ¢ 3 ¢ XF ')( $ 5_7 G F]3 G a 1 -21
7 5n')( $ 5 7 ¤
' % &-21 J ( L ']")"$&% ' ( $ ! H - & 3 ¢ &
7 &-21 43 ¢ 43 ¢ XF ')( $ 5_7 G F G a 1 -21 7
5n')( $ 5 7 ¤
and keiν(z)
1.17.1. Derivatives with respect to the argument
1. f;/ 0 F m G I+ -; . D ' 2 0f;/ <3
' 2 0f;/ H ¤ 2. f;/ 0
F m G + -; . D ' 2 0f;/ ' 2 0f;/ H ¤
3. I+ a 1 . a 1 f / 0 3 7 F m G &-21 Y D O')(:7X! FOf " G FOf " G ')(:7! F f " PG F f " G9H ¤
4. I+ a 1 . a 1 0f;/ 7 F m G &-21 Y D O')(87! FOf " PG F f " PG 3
')(:7! F f " G F f " G9H ¤ 5. I+ a 1 . -;&-21 0f / ( 7! ' & F m G &-21 Y D O')(87! FOf " G F f " G3
')(:7X! F f " G F f " G9H ¤ 6. I+ a 1 . -;&-21 0f;/ ( 7X! ' F m G &-21 Y D ')(:7X! F f " G FOf " PG
O')(:7X! F f " G F f " G9H ¤ 7. k - + a & . a 1 0f / / F mO G a 1 Y O'65_7X! D a 1 F m G -;&-21 F m G
3 a 1 F m G -;&-21 F m G9H 5L
# " "! #$" #$$! 3
O'65 7! D a 1 F m G -;&-21 F m G a 1 F m G -I&-21 F m GIH ¤ 8. k - + a & . a 1 f;/ / 8F mOG a 1 Y '65_7! D a 1 F m G -I&-21 F m G3 a 1 F m G -;&-21 F m G9H
'65_7! D a 1 F m G -;&-21 F m G a 1 F m G -I&-21 F m GIH ¤ 9. EDM &-21 A F m &-21 0fI& m &-21 fI& G9H
m ! '#" &-21 D O')(87! / £ fI H = '8>?7A@ B 10. ED &-21 A F m &-21 0fI&3 m &-21 fI& G9H
m ! '#" &-21 D O')(87! / £ fI H = '8>?7A@ B 11. D &-21 F m m &-21 fI&3 m m &-21 fI& G9H
m ! '#" &-21 D ' (87! / £ fI / £ fI 3
O')(87! / £ fI / £ fI H ¤ 12. D &-21 F m m &-21 fI& m m &-21 fI& G9H
m ! ']" &-21 D O')(:7X! / £ fI / £ fI O')(87! / £ fI / £ fI H ¤
13. k &-21 <1 f / <3 <1 f / m ']" I+ &-!& . Y D O')(:7X! &-21 £ f;/ <3
O')(:7X! &-21 £ f;/ H= '*>?7R@CB 5P
+ ( !< 14. k &-21 <1 0f / <1 f /
m ']" I+ &-!& . Y D O')(87! &-21 £ f / O')(:7X! &-21 £ f / H= '*>?7R@CB
15. 0f;/ F m G I+ -; . 2D ' 2 0f / 3
' 2 0f / H ¤ 16. 0f;/
F m G I+ -; . 2D ' 2 f;/ ' 2 0f;/ H ¤
17. + a 1 . a 1 0f;/ 43 ¢ ' & f &-21 - / A Df " O'65 ! H ¤
18. I+ a 1 . a 1 0f;/ 43 ¢ a 1 '%& f &-21 - / A D f " O'65 ! H ¤
19. k - + a & . a 1 f;/ ( 7! ' F mO G a 1 Y O'65_7X! D a 1 F m G 3 a 1 F m G9H 3 £ '65_7X! a 1 F m G a 1 F m G ¤
20. k - + a & . a 1 f;/ ( 7X! ' F m G a 1 Y '65_7! D a 1 F m G 3 a 1 F m GIH £ '65_7X! a 1 F m G a 1 F m G ¤ 21. k &-21 &-21 f / &-21 0f / 3 ¢ f &-21 I+ &-!& . Y D ')(:7! &-21 £ f;/ <3
O')(:7X! &-21 £ f;/ H ¤ 22. k &-21 &-21 0f;/ 3 &-21 0f;/
43 ¢ / f &-21 I+ &-!& . Y D ')(:7! &-21 £ f;/ O')(:7X! &-21 £ f;/ H ¤
;9N
# " "! #$" #$$! 23. k &-21 &-21 0f;/ &-21 f;/ 3 &-21 0f;/ &-21 f;/ 4
m '#" I+ &-!& . D O')(:7X! &-21 £ f / 3
O')(87! &-21 £ f;/ H ¤ 24. k &-21 &-21 0f;/ &-21 f;/ &-21 0f;/ &-21 0f;/ 4
m ']" + &-!& . D ')(:7! &-21 £ f;/ O')(:7X! &-21 £ f / H ¤
1.17.2. Derivatives with respect to the order
1.
M! 3 0&3 & ' % &-21 J L ""'$9% ')( $ ! D ] $ (*'! 0& # $ (*'! & H ¤
2.
# $ (*'! & H ¤ 3.
$ (*'! & H ¤ 6.
;
7#7 ! ( 1
1 & & & #$
1 1 & #$
& ! ( 1
')( $ ! a 1 & H 3 ' % 1 J L
"')( $ ! % $ -21
% &- a 1 0&Y D $ (* (:7X! -21 £ & $ (*Q(:7X! -21 £ & H &- a 1 &&D $ (* (:7! -21 £ &3
$ (* (:7! -21 £ & H3 43 ¢ a a -;&-21 0&D $ (*Q(87! 1 - £ & $ (*Q(:7X! 1 - £ & H3 3 ¢ a a -;&-21 &&D $ (*Q(:7X! 1 - £ &3
$ (* (:7! 1 - £ & H = @ B 8.
M! a 1 M! a 1
7 a 1 & # 1 N 1 ! (
1 & & & #$
1 !
& ! ( 1
')( $ ! a 1 & H ' % 1 J L
"')( $ ! % $ -21
% &- a 1 0&Y D $ ( Q(87! -21 £ &<3 $ ( (:7! -21 £ & H3 &- a 1 0&D $ (*Q(:7X! -21 £ &
$ (* (:7! -21 £ & H3 3 ¢ a a -;&-21 &&D $ (*Q(:7X! 1 - £ &3 $ (*Q(:7X! 1 - £ & H 43 ¢ a a -;&-21 &&D $ (*Q(87! 1 - £ &
$ (* (:7X! 1 - £ & H = @ @ B 9.
M! 1 -; 7
1 & & & #$
1 1 & #$
& ! ( 1
3 ' % &-21 J ( L ""'$9% ')( $ ! D ')( $ ! 1 - & ')( $ ! 1 - 0& H
;93
"')( $ ! % $ -21
% 3 ¢ a &- -21 0&Y D $ (*Q(:7X! 1 - £ & $ ( (:7! 1 - £ & H 3 ¢ a &- -21 &&D $ (* (:7! 1 - £ &3
$ (*Q(:7X! 1 - £ & H3 43 ¢ -; a 1 &&D $ (*Q(87! -21 £ & $ (* (:7! -21 £ & H3 43 ¢ -; a 1 0&D $ (*Q(:7X! -21 £ &3
$ (*Q(:7X! -21 £ & H = @ @ B 10.
M]! 1 -; 7 1 -; 0&
V 1 N 1 ! (
3 3 ¢ / £ &-21 0&3 &-21 0& 1 N 1 ! (
1 1 & #$
& ! ( 1
' % &-21 J ( L ""'$9% ')( $ ! D ')( $ ! 1 - &<3 ')( $ ! 1 - 0& H
3 ' % 1 J L "')( $ ! % $ -21
% 3 ¢ a &- -21 0&Y D $ ( Q(87! 1 - £ &<3
$ ( (:7! 1 - £ & H3 43 ¢ a &- -21 0& D $ (*Q(:7X! 1 - £ & $ (* (:7! 1 - £ & H3 3 ¢ -; a 1 & D $ (*Q(:7X! -21 £ &3 $ (*Q(:7X! -21 0& H
;!5
$ (*Q(:7X! -21 £ & H = @ @ B 11.
M]! &-21 &-21 0&3 &-21 &3 £ & &-21 & 43 ¢ C £ & 1 -; 0&3 3 ¢ 1 -; & 3 &-21 0& 43 ¢ 1 -; &0 1 N
1 ! ( 1 &
& &
#$
&-21 0& &-21 0& 43 ¢ 1 -; 0& 3?43 ¢ 1 -; &0 1 N
1 !
3 ¢ 1 -; & 3 ¢ 1 -; 0& 1 N & ! (
1
7 &-21 0&3 43 ¢ 1 -; &0 N 7#7 ! (
1
3 ' % 1 J ( L
"')( $ ! % $ -21
% &- -21 0& 43 ¢ a -; a 1 & Y D $ 5 P5 ! -21 £ &<3 $ 5 P5 ! -21 £ & H 3 ¢ a -; a 1 0&3 &- -21 0& Y D $ 5 P5 ! -21 £ &
$ 5 P5 ! -21 £ & H = @CB 12.
M! &-21 3 &-21 & C £ & 0 &-21 0& 43 ¢ 1 -; 0& 43 ¢ 1 -; 0& ; ;
+ ( L1 3 &-21 0& 43 ¢ 1 -; &<3 &-21 &0 1 N
1 ! ( 1 & & &
#$
3 &-21 0&3 &-21 0& 43 ¢ 1 -; 0& 43 ¢ 1 -; &0 1 N 1 ! (
1 1 & #$
& &-21 & &-21 &3 3 ¢ 1 -; 0& 43 ¢ 1 -; &0 1 N
& ! ( 1
1
3 ' % 1 J L "')( $ ! % $ -21
% Y 43 ¢ &- -21 & 3 ¢ -; a 1 & Y D $ 5 5 ! -21 £ &
$ 5 5 ! -21 £ & H M3 ¢ &- -21 0&3 43 ¢ -; a 1 0& Y D $ 5 P5 ! -21 £ &<3 $ 5 P5 ! -21 £ & H = @CB
13.
M]! 1 -; 43 ¢ a 1 &-21 0&! 3 ¢ a 1 M]! &-21 ¤ 14.
M! 1 -; 43 ¢ a 1 &-21 0& 43 ¢ M! &-21 ¤ 1.18. The Legendre Polynomials Pn(z)
1.18.1. Derivatives with respect to the argument
1. j S 0fI& £ 3 ¢ f a 1 S -; fI& = > 'I@CB 2.
5n'! % J 1 L ' (*'! % F m G ¢ 3hf9 O -; 1 -;SEa fI& = > 'I@CB
;9:
& ' 3. -21 ¢ 3hf & &-21 j f;/ 0
43 ¢ F 7 G f 63hf -;&-21 j F 7m G ¤ 4. DM Q3hf &-21 j F m GIH
43 ¢ F 7G f9 Q3hf9 O -;&-21 j F m G ¤ 5. a 1 ¢ 3hf9 & &-21 j a 1 f / 0
3 ¢ O'65_7 F 7 G f9 a 1 -;&-21 ¢ 3hf9 O& -;&-!&4 j F 7m G ¤ 6. D - + a 1 . Q3hfI &-21 j F 5 m m G9H
F 7 G f -;&-21 63hfI -21 j F " m G ¤ 7. D fg3h& &-21 j F 5 m m G9H
43 ¢ F 7 G f fT3h& -21 j F" m G ¤ 8. DM S fg3h& &- S -21 j S F 5 m m GIH
% ( '2! % f + S -; . 0fg3h& - S -21 j S -; F 5 m m G = > 'I@ B 9. DM - + SEa 1 . 0fg3h& SEa j S F 5 m m GIH
43 ¢ 5n'! % % f - + SEa a 1 . fg3h& S j SEa F 5 m m G ¤ 10. D 0Q3hfI 2j F 5 m m GIH f D j F" m G9H ¤ 11. D 0 3hfI& S j S F E( m ( m GIH
( ! "'C7 ( ! ' ! % (* '! % 0 3hfI& + S -; . j S -; F O( m ( m G = > 'I@CB 12. D &- S -21 0fg3h& S j S F m (8 m ( m GIH " ' m '
J 1 ( L' ! % ( O'! % - S -21 fg3h& + S -; . j S -; F m (8 m ( m G = > 'I@CB 13. ED 0 3hf - + SEa 1 . j S F ( m GIH
43 ¢ 5n'! % % 3hf9 - + SEa a 1 . j SEa F ( m G ¤ ; <
+ ( L1 14. ED 0fg3h& -21 j F#" ¢ 3 m GIH
F 7 G 0fg3h& -;&-21 j )* ( m, ¤ 15. a 1 D a 1 j a 1 F " ¢ 3 m GIH
3 ¢ 'g5 7 F 7 G 63hf9 -;&-21 j )+* E( m-, ¤ 16. k &-21 0fI)3 ¢ -21 j / ¢ 3 fI
F 7 G -;&-21 fI 3 ¢ -I&-21 j F 7 7E( m G ¤ 17. D &-21 ¢ 3hfI& j F 7 7( m GIH
3 ¢ F 7 G -;&-21 j / ¢ 3hfI ¤ 18. ED 0 3hf S j S F ( m GIH
! % ( O'! % 3hf9 S -; j S - F ( m G = > 'I@ B 19. a 1;D 0 3hf9 O S j S:F ( m GIH
! % ( O')(:7X! % 3hf S -;&-21 j S - &-21 F ( m G = > 'g5 7R@CB 20. ED -;&-21 0f9 3h j F m m (8 GIH
43V F 7 G f - &-21 f9 3h j F m m (8 G ¤ 21. D &- S -21 0f 3h S j S F m m (8 GIH
! % (* '! % f9 - S -21 0f9 3h S -; j S - F m m (8 G = > 'I@CB 22. a 1;D &- S 0f9 3h O S j S:F m m (8 GIH 3 ! % (* ')(:7X! % f9 a 1 - S -21 0f9 3h S -;&-21 j S - &-21 F m m (8 G= > '65_7R@CB
23. &-21 ¢ 3 f9 & a 1 < ¢ 3hf9 & &- S -21 j S 0f;/ ! % (* '! % F m G -21 ¢ 3hf9 & - S -21 j S - 0f;/ = > 'I@CB
;9L
) ' * ' 24. ED 0Q3hf9 a 1 ED S 0Q3hf9 &- S -21 j S:F m GIHOH
! % (* '! % F m G S -; 63hf9 - S -21 j S - F m G = > 'I@CB 25. D P D fT3h& S j S F m 5nm (* G9HOH
D % (*'! % H fg3h& S -; j S -; F m 5 m (8 G = > 'I@CB 26. D P D SEa fg3h& - S -21Oj S F m 5 m (8 GIHOH
D 5n'! % % H f S fg3h& - S -;&-21 j SEa F m 5 m (8G ¤ 27. ED PED fT3h& - S -21 j S F m 5 m (8GIHOH
D 5:'! % % H fg3h& - S -;&-21Oj SEa F m 5 m (8 G ¤ 1.19. The Chebyshev Polynomials Tn(z) and Un(z)
1.19.1. Derivatives with respect to the argument
1. ST fI&0 £ &-21 3 ¢ f S -; 0fI& = > '8>?7A@CB 2. EDM -1 ] fg3h& F m 5 m (8GIH 3 ¢ F 7G -;&-21 0fg3h& ¤ 3. 0f9 3h S S ) m 5 m (8 ,
! % ( O'! % f9 3h S -; S -; ) m 5 m (8 , = > 'I@ B 4. &- S -21 0 3hf9 S S ) 5 m ( m , ! % ( O'! % f9 Y - S -21 3 f9 S -; S -; ) 5 m ( m , = > 'I@CB 5. D0 f 3h S S F m m (8 GIH
! % (* '! % 0f 3h S -; S - F m m (8 G = > 'I@ B 6. a 1;D0 f9 3h S S:F m m (8 GIH
! % (*O')(:7X! % ; f9 3h S -;&-21 S - &- F m m (* G = > 'g5 7R@CB ;9P
+ ( P1 7. ED0 f9 3h SEa 1 SEa 1 F m m (8 GIH
5_7! % (* 'g5 7! % f9 3h O S -; a 1 S - a 1 F m m (8 G = > 'I@CB 8. ED0 3hf9 O - S S:F ( m G9H
3 5n')(:7X! % (:7X! % 3hf9 - + SEa . SEa F ( m G = 5n'*>?7R@CB 9. EDM &- S -21 3hf9 S S F ( m GIH
! % (*O'! % f9 - S -21 3 f9 S -; S - F ( m G = > 'I@CB 10. a 1;D &- S 0 3hf9 O S S F ( m GIH 3 ! % (* ' (87! % f9 a - S -21 3hf9 S -;&-21 S - &- F ( m G= > '65_7R@CB
11. ED &- S - 0 3hf9 SEa 1 SEa 1 F ( m GIH 5_7X! % ( O'65_7X! % f9 - S - 0 3hf9 S -; a 1 S - a 1 F ( m G= > 'I@ B
12. ED &-21 PED fT3h& S S:F m 5 m (8GIHOH £ - ! % (*O'! % -21 ] fg3h& S -; S -; F m 5nm (* G = > 'I@ B
13. D a 1 D &- S -21 fg3 S S F m 5 m (8 GIHOH ! % (* '! % F m G - S -21 fg3h& S -; S -; F m 5 m (8G = >_'I@CB
14. ED a 1 PED -21 0fg3h& - S S8F m 5 m (8GIHOH £ - £ fT3h - S -; SEa F m 5 m (8 G ¤
15. ED &-21 PED SEa &-21 fT3h& - S S F m 5 m (8 GIHOH £ - £ f S -21 0fg3h& - S -; SEa F m 5 m (8 G ¤
16. ST0fI& £ fI a 1S -; 0fI& = > 'I@CB : N
Tn(z)
Un(z)
17. ; f9 3h S S ) m 5n m (* , 5n ! % (* '65n ! % ;0f9 3h S -; S -; ) m 5 m (8 , = > 'I@ B
18. &- S - f9 3h S S ) m 5n m (* , 5nO! % (* '65nO! % f - S - 0f 3h S -; S -; ) m 5 m (8 , = > 'I@CB
19. D;0f 3h S S F m m (8 G9H 5_7! % (*O'65 7! % ; f9 3h O S -; S - F m m (* G = > 'I@ B
20. ED;0f9 3h SEa 1 SEa 1 F m m (* G9H 5:O! % (*O'65n ! % ; f9 3h S
H A N D B O O K O F
Yury A. Brychkov Computing Center of the Russian
Academy of Sciences Moscow, Russia
Special Functions Derivatives, Integrals, Series
and Other Formulas
H A N D B O O K O F
Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742
© 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1
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Contents
+**,!"! '
+**,!"!" '
1.4. The Sine si (z) and Cosine ci (z) Integrals ( # +**,!"! (
1.5. The Error Functions erf (z) and erfc (z) -
' +**,!"! -
& +**,!"! .
( +**,!"! .
- +**,!"! - +**,!"!"
/ +**,!"!
. +**,!"! '
+**,!"! /
1.12. The Hankel Functions H (1) ν (z) and H
(2) ν (z)
+**,!"!
# +**,!"! & # +**,!"!0 #.
1.16. The Anger Jν (z) and Weber Eν(z) Functions # '
& +**,!"! #'
& +**,!"!0 #(
( +**,!"! #-
- +**,!"! '&
/ +**,!"! '/
. +**,!"! &
+**,!"! &
+**,!"!" &#
+**,!"! &# +**,!"!" &&
+**,!"! && +**,!"" &/
# +**,!"! &/
.
' +**,!"! (.
' +**,!"" (
1F1(a; b; z) (
1.28. The Whittaker Functions Mµ,ν(z) and Wµ,ν(z) -.
- +**,!"! -.
/ +**,!"! -.
/ +**,!"" -'
. +**,!"! -& . +**,!"" -(
4.3. The Sine Si (z) and Cosine ci (z) Integrals -
# )0 - # )0 - # )0!! -
#' )"00 -#
4.4. The Error Functions erf (z), erfi (z) and erfc (z) -#
## )0 -# ## )0!" -'
4.5. The Fresnel Integrals S(z) and C(z) -/
#' ) 0 -/
#' ) 0 /.
#& )0 /
#& )0!" /
#& )0 /
1
#( )0 /& #( )0!" /( #( )0 /( #(# )0!! //
#- )0 .# #- )0 .#
#/ ) 0 .'
#. ) 0 & #. )"0 &
# )H L
0 ( # )H
0*
4.12. The Kelvin Functions berν(z), beiν (z), kerν(z) and keiν (z)
# ) 0
# )"00
## )0
#' )0 # #' )0 & #' )0" #
#& )0 # #& )0 #
#& )0 #& #&# )"0 #&
#( )0 #( #( )0!" #-
'# #- )
'( #/ )
&
# )K 0 &'
# )K !"!%"0 &(
# )K 0!! &-
## )K 0* (
#' )K 0 (#
# )E 0 (/
# )E !"!%"0 -
# )E 0!! -&
## )E 0* -/
#' )E 0 /
# )D 0% ..
## )0 .
' ./ ' "0 '
5.2. The Incomplete Gamma Functions γ(ν, z) and Γ(ν, z)
' ' "0 '
' # ' "0 #
'# ' '# "00 ' '# "0 (
'' - '' 0" - '' "0 /
'& H 0L
'( . '( 0" ' '( "0 ' '(# & '(' - '(& "0 -
'- / '- "0 '- '-# '-' "0 ' '-& '
'/ ( '/ 0" #. '/ "0 # '/# # '/' # '/& "0 ##
## '.
##
'.
#& '.
#-
'.#
& '
& '
&' '
&'
'#
/ '
!
/ '
!
/
' !
'
#
1F1(a; b; z) #'# # '# 0" # '# "0 #
5.15. The Tricomi Confluent Hypergeometric Function Ψ(a; b; z) # '' # '' 0" ##
5.16. The Gauss Hypergeometric Function 2F1(a, b; c; z) ##'& ##
'( #- '( 0" # '( # '(# 0" ##'(' "0 #' '(& #'
5.18. Multiple Sums # &
& # & 0 ##( & "0 ##( &# ##/
& #'
&# # #' &# # #'
&' # #0 #'
&' # #0 #''
&' # #0 # #'-
&& #'/
&( 0% #&.
&- ! #&'
&/ #('
&. H 0L
& #-& & 0 #-& & "0 #--
& #-/
/'
&#
#/'
&#
//
&'
#//
&'
'.'
&& !
'.'
&& !
&( '.-
8.1. The Hypergeometric Functions '&
m,n p,q
' - % &' - " &'' &&/ )0 0 &( )0% &(/
1
Preface
1.1. Elementary Functions
1.1.1. General formulas
1. ! '#" "$&% ')(* $ ! % ,+ &- . ¤ 2. 0/ 0 &-21 43 ¢ '65 $ (87! %$9% ')( $ (:7! % £ / -;&- <+ &- . / = '8>?7A@CB 3. ED EF 7GIH 3 ¢ 3 ¢ &-21 KJ ' $L "" '')( $ (:7! % ,+ &- . F 7G = '8>?7A@CB 4. EDM &-21 EF 7 GIH 3 ¢ -;&-21ON F 7 G < PQ , & N & ¤ 5. 1 &R 0& ¤¤4¤ ST0& U4 J '$ LV + U4.1 & &- UXW4 J ')( $ $ LV + XW4. 0& ¤¤4¤
Y &- U -<Z Z Z - X[<\]W [<\#U ^ ')( $ (_BB B<( $ ( " $ ( " ` + [<\#U .S -21 &R + &- U -<Z Z Z - X[<\#U4.S & ¤ 6. EDM SEa F 7 G S D S -21Ab F 7 G9HOH 43 ¢ SEa -;&-21 N F 7 G
&9 S b 0& N 0& ¤ 1.1.2. Algebraic functions
, 5. c f & c F m G ( &! '
J 1 ( L ' c -; 0f & c -; c -; a 1 F ¢ m G ¤ 6.
3 ¢ 0 fI& c -; - c F O 5 m 5 m G ¤ 7. -21 0f & -21
£ 3 ¢ a 1 f -21 f & -;&-21 ¢ 3 F m 5 G + a 1 . a 1 F m 5nO m 5 G! ¤ 8. - c -21 ] fg3h& c
£ - '! % J 1 ( L ' - c -21 0fg3h& c -; c -; a 1 F#" m G ¤
9. c fg3h& &- c -21 43 ¢ f c -; 0fg3h& - c -21$ - c F 5 m m G ¤ 10. &- c -21 0f & c
F3 7 G (9! ' J 1 (% L ' &- c -21 f & c -; c -; a 1 F ¢ m8G ¤
11. - c -!&4 0fg3h& c 3 ' % J & L 'J ('6( 1 L '%& f -21 O - c -21 fg3h& c -; c -; a 1 a 1 F#" m G ¤
12. &-21 0fg3h& c J 1 L 'J 1 ( L ' f -21 0fg3h& c -; c -; a 1 F " m G ¤ 13. a 1 0f & c
,
19. 0f & -;&-!&4 f -21 ] f- & -;&-21Oj a 1 )+* mm 5 , ¤ 20. 0f & f j F ¢ m G ¤ 21. -;&-21 0f & 3 ¢ -;&-21 j F ¢ m G ¤ 22. &-21 fh& -;&-21 43 ¢ -21 0f h& -;&-21 j )+* m 5: , ¤ 23. &-21 0f & 3PfI -21 j F" m G ¤ 24. a 1 0f & a 1 J & L ''g5 7 f 1 0f & 1 F ¢ m G ¤ 25.
J & L '#'Q5_7! f a 1 1 a 1 F" ¢ m G ¤ 26. &-21 0f & a 1 F 7 G f a 1 -21 a 1 F#" ¢ m G ¤ 27. a 1 0f & &-21 F 7 G f 1 0f & -21 F " ¢ m G ¤ 28. a 1 0fg3h& &-21 43 ¢ F 7G f a 1 fg3h& -21 a 1 F#" m G ¤ 29. &-21 0f & &-21 F 7G f -21 f- & -21 F#" ¢ m G ¤ 30. &-21 0f & -; F 7G -21 f & -; )+* mm 5 , ¤ 31. a 1 0f & -;&-21 F 7 G 1 f- & -;&-21 )+* mm 5n , ¤ 32.
43 ¢ F 7G f- & -;&-21 a 1 ) * m 5 , ¤ 33. &-21 0f & -;&-21
F 7G fI& -21 0f & -;&-21 a 1 ) * mm 5 , ¤ 34.
43 ¢ F 7 G -21 0fh -;&-21 )+* m 5: , ¤ 35. a 1 0f & -;&-
43 ¢ £ - O'65_7X! %'65 7! % 1 f- & -;&- F ( m 5 m G ¤ 3
3 36.
J & L '#'Q5_7! F m G 1 f & -;&-!&4 a 1 )+* mm 5n , ¤ 37. 0f9 3h c 3 £ fI ('9! '' ( &! ' 0f9 3h c -; c -; a 1 F m G ¤ 38. 0f9 3h -21
f -21 fg3h& -;&-21 ¢ 3 F E( m 5 m G + a 1 . a 1 F ( m G ¤ 39. 0f9 3h -21 £ f -21 f9 3h O -;&-21 a 1 F m m (8 G ¤ 40. a 1 0f9 3h -21 £ ¢ f -21 ; f9 3h O -;&-!&4 a 1 F m m (8 G ¤ 41. -21 0f & -21
43 ¢ - + a 1 . 0f & - + a 1 . j F 5 m m 5 G ¤ 42. 0f9 -21 R 43 ¢ f9 - + a 1 . j F m 5 G ¤ 43. 0f9 3h 43 £ fI j F m G ¤ 44. 0f9 3h &-21 R 43 £ fI F 7 G f9 3h -21 F m G ¤ 45. 0fIQ3h &-21 R 43fI F 7 G 0fIQ3h -21 F" m G ¤ 46. 0f 3h a 1 ( m ! ''65_7 F G 0f 3h 1 F m G ¤ 47. -;&-21 0f9 3h 43 £ fI -;&-21 j F m G ¤ 48. &- c -21 0 3hf9 c
£ fI ('9! '')( &! ' &- c -21 0 3hf c -; c -; a 1 F m G ¤ 49. 7 5 m F $ f G 1
3 ¢ m F 7 G 0 f9 - + a 1 . F O'65_7 m G ¤
1.1.3. The exponential function
1. k c - c -; - c -; 0fI& ¤ 2. c -R 43 ¢ c -; -R - c -21 F m G ¤
5
3. ED W H 3A f W R/ f& ¤ 4. EDM &-21 R W H f -;&-21 R W F m G ¤ 5. ED - W H 3 ¢ f - W / f& ¤ 6. EDM &-21 #-R W H f -;&-21 #-R W F m G ¤ 7. / 8F m G a 1 + 1A- . &-21 f;/ 1 -; 0f;/ ¤ 8. D 7 H / *F m G a 1 - + 1 a . a 1 f;/ -;&-21 f;/ 0 ¤ 9. k &-21 R
3 ¢ / *F mG a 1 - + a . D &-21 F m G 1 -; F m GIH ¤ 10. k &-21 R
43 ¢ / F m G a 1 - + & a . D a 1 F m G -;&-21 F m GIH ¤ 11. - 3 ¢ m ' & ']" I+ 1A- . &-21 f / ¤ 12. D 7 - H 43 ¢ m '%& '#" - + 1 a . a 1 0f / ¤ 13. &-21 -R m '%& ']" - + a . &-21 F m G ¤ 14. k -;&-21 #-R m '%& ']" - + & a . a 1 F m G ¤ 15. D + -21 . U H 3 ¢ a 1 R/ *F m G a 1 + 1A- . + .1 -; f;/ = 7@CB 16.
43 ¢ / 8F m G a 1 + 1A- . + .&-21 0f;/ = 7]X@CB 17. D 7 + -21 . U H
3 ¢ a a 1 R/ *F m G a 1 - + a 1 . + . a 1 0f;/ = 7@CB 18.
/ F m G a 1 - + a 1 . + .-;&-21 0f / = 7]X@CB ;
5 1.1.4. Hyperbolic functions
1. &c fI&0 ' % &c -; c -; 43fI&<3 #- c -; fI& ¤ 2. c fI&0 ' % c -; c -; 43fI& - c -; fI& ¤ 3. fI&0
£ 3fI ")" $ 5 7! " " 5_7! 3 "'& 8 S 43 ¢ S J $ L ¤
#$ ¤
#$ ¤
12. D & H £ 43 ¢ F 5 G a 1
:
13. ED H 43 ¢ a 1 ' % '%&
# ( 7! ' ' % ' & a & N a 7( &B B B ( &B B B 7 ( 9BB B #7( #7,5 &B B B #7<5 ! 7 #$ ¤
14. D &-21 F m GIH 3 ¢ / F m G a 1 - + a . 1 -; F m G ¤ 15. ED &-21 F m GIH 3 ¢ / *F m G a 1 - + a . &-21 F m G ¤ 16. D &-21 F m GIH 3 ¢ / F m G a 1 - + & a . a 1 F m G ¤ 17. ED &-21 F m GIH 3 ¢ / F m G a 1 - + & a . -;&-21 F m G ¤ 18. 0f / '%& f a 1 I+ 1A- . &-21 '65 $ !$&% ')( $ ! 43fI - - -; a 1 0f / = '*>?7R@CB
19. 0f / ' & f a 1 I+ 1A- . &-21 '65 $ !$9% ')( $ ! 43fI - - &- -21 f / = '*>?7R@CB
20. m !
' & f a 1 - + a 1 . &-21 '65 $ !$9% ')( $ ! 3PfI - - &- a 1 f / = '*>?7R@CB 21. m !
'%& f a 1 - + a 1 . &-21 '65 $ !$&% ' ( $ ! 43fI - - -;&-21 0f / = '*>?7R@CB
1.1.5. Trigonometric functions
1. 0fI& f F fI ' G ¤ <
; 2. 0fI& f F fI ' G ¤ 3. 0fI& 43 ¢ + a 1 . f 1
( 7! " " J '65_7$ 5 7 L a 1 fI& Y S J $ L 3 £ D 3 £ fIQ3 7( ( 7! ' H = = @ !C@CB
4. 3 ¢ + a 1 . f 1
( 7X! " " J '65_7$ 5_7 L S J $ L 3 £ SEa 1 fI& Y + - . - S
3 ¢ J $ (* 5 L a fI& '
! = @O !#B 5.
3 ¢ + a 1 . f fI& 7 " + - . S 43 ¢ S J $ 5 L Y SEa 0fI&
43 ¢ J $ L £L ¢ '
! = @ !#B 6.
£ AfI " $ 5_7X! 5 7! 3 "'& 8 S 3 ¢ S J $ L = = @ !C@CB
7. 3 £ a 1 m ' ' % m 5 ! '%& m ' ! '%& D + . F 5n m G 3 + . F 5n m G9H
3 43 ¢ m ' ! '%& D + . F (* m G 3 + . F 3 5n m GIH ¤ 8. fI&0
3 ¢ f 1 ( 7! " " J '65_7$ 5_7 L S J $ L 3 £ SEa 1 fI&
Y + - . - S 3 ¢ J $ (* 5 L a 0fI&
' ! = @ C7X O! B
7 " S 43 ¢ S J $ L £ ¢ Y + - .
3 ¢ J $P5 L a 0fI& '
! = @ C7 O! B L
10.
43 ¢ a 1 ' %m ' & m ' ! '%& D + . F 5 m G 3 + . F m G9H 3 43 ¢ m ' ! ' & D + . F ( m G 3 + . F3 m G9H ¤
11. 0fI& 43 ¢ a 1 £ fI 1 $"'& m ]! "
Y D 3 ¢ 4fI$ 7( ( 7! ' H S 1 3 ¢ S J $ L = = @ C7 !C@CB 12.
43 ¢ 4 a 1 £ fI fI& 1 7 " S 1 3 ¢ S J $ L
Y + a V- . 3 ¢ J $ (87Q( Q5_7 L - a 1 fI&
' ! = @ ! B
13. m '
' & D + . F 7 m G 3 3 ¢ + . F 7 3 m GIH ¤ 14. fI&0 43 ¢ + &-21 . C £ fI fI& 7 " S 1 3 ¢ S J $ L
Y + a V- . 43 ¢ J $ (:7Q( 5_7 L - a 1 fI&
' ! = @ ! B
15. m '
' & D 3 ¢ + . F]3 m G 3 + . F ¢ m G9H ¤ 16. c fI&0 ' % c -;
c -; 43 AfI&3 - c -; AfI& ¤
17. c 0fI& ' % c -; c -; 43 AfI& - c -; AfI& ¤
18. fI 0 ( 7X! ' f + &- . D W & / fE3 - W / f H ¤
19. 0fI ( 7X! ' f D W & / f - W / f H ¤
20. ED &-21 m H 3PfI -;&-21 F m ' G ¤ 21. D &-21 m H 3PfI -;&-21 F m ' G ¤
P
; 22. f / 0 / *F m G a 1 I+ 1A- . 1 -; f / ¤ 23. 0f;/ 3 ¢ / *F m G a 1 I+ 1A- . &-21 0f;/ ¤ 24. S 0f;/ <
#$ ¤
#$ ¤
30. ED H £ #F ( G a 1
Y a & N a 7 1 ( 9BB B 1 ( 1 5 &B B B 1 5 & ( &B B B & ( & 5 &B BB & 5 ! 7 #$ ¤
31. D H £ 43 ¢ -;&-21 Y a & N a 7 V( 9BB B V( &B B B ! 77 ( &B BB #7( #7<5 &B B B V7<5
#$ 3 43 ¢ ' % '%& ¤ N
32. ED &-21 m H 43 ¢ / 8F mG a 1 - + a . 1 -; F m G ¤ 33. ED &-21 m H / 8F mG a 1 - + a & . a 1 F m G ¤ 34. D &-21 m H / F m G a 1 - + a . &-21 F m G ¤ 35. ED &-21 m H 43 ¢ / 8F mG a 1 - + a & . -;&-21 F m G ¤ 36. f / 0 £ - &-21 O/ f a 1 I+ 1A- .
Y &-21 43fI - '65 $ !$9% ')( $ ! - -; a 1 f / = '8>?7A@ B 37. 0f / 3 ¢ £ - &-21 / f a 1 I+ 1A- .
Y &-21 f - '65 $ !$9% ')( $ ! - &- -21 f / = '8>?7A@CB 38. m !
43 ¢ £ - &-21 / f a 1 O - + 1 a .
Y &-21 f - '65 $ !$9% ')( $ ! - &- a 1 f / = '8>?7A@CB 39. m !
£ - &-21 / f a 1 - + 1 a .
Y &-21 43 ¢ f - '65 $ !$&% ')( $ ! - -;&-21 0f / = '8>?7A@ B 40. RED
H 3# ¤
42. R a D H 643 ¢ £ a 1
¤ 43. R a & D
H 3V ¤
47. R a & D H 3 ¢ a 1 £ a 1 & ¤
48. f;/ 0f;/ f;/ ¢ / 8F m G a 1 I+ 1A- . Y D O'65 ! <1 f / £ '
'Q5 ! <1 f / £ H ¤ 49. f;/ 0f;/
f;/ ¢ a 1 / *F m G a 1 I+ 1A- . Y D O'65 ! 2 1 f / £ K3 O'65 ! 2 1 f / £ H ¤
50. 7 f;/ 0f;/ f;/ ¢ / *F m G a 1 - + a 1 . Y D O'65 ! 2<1 f / £ '
'Q5 ! 2<1 f / £ H ¤ 51. 7 0f / f;/
f;/ ¢ a 1 / *F m G a 1 - + a 1 . Y D O'65 ! 1 f / £ E3
'Q5 ! 1 f / £ H ¤ 1.1.6. The logarithmic function
1. / / f 43 ¢ &-21 ')(:7X! % -; f9 -; j &-21 ) 5 m 5 m , = '*>?7R@CB
2. k &-21 / f / f ')(87! %O 3 ')(87! %O m 5 m j &-21 F 5n m m 5 m G = '*>?7R@CB
3. D F $ f G9H 43 ¢ &-21 3 ¢ 0 f9 O -; j &-21 F 5 m G = '*>?7R@CB
4. D &-21 F f f GIH ')(87! % 3 ')(:7X! % f f9 -; j &-21 F m 5 m G = '*>?7R@CB
5. ED m 5 m (8<H £ £ 3 ¢ ;0f9 3h -;&-21 &-21 F m m (* G = '8>?7A@CB !,
6. a 1;D m 5 m (8<H £ £ 0f9 3h O -;&-21 a 1 F m m (8 G ¤ 7. D m 5 m ( H
3 ¢ f &-21 1 -; f 3h& -; j + 1 -;9lC-; .&-21 F ¢ 3 m G = '*>?7R@CB 8. D &-21 m 5 m ( H
43 ¢ 3 ¢ fI &-!&4 ] Q3hf9 O -; j + 1 -;9lC-; .&-21 ) ¢ 3 m , = '*>?7R@CB 9. EDM 1 f9 3h& EDM &-21 m 5 m ( H H ¡ = '8>?7A@CB 10. ED &-21 0f9 3h& ED m 5 m ( HOH ¡ = '8>?7A@CB 11. ED -21 0f9 3h& ED &-21 m 5 m ( HOH
£ ) 3 m , -;&-21 m 5 m ( ¤ 12. D a 1 0f 3h& D -21 m 5 m ( HOH '! % ' m 5 m ( ¤ 13. ED a 1 PED -21 0f 3h& &-21 m 5 m ( HOH
43 ¢ F 7 G f f 3h& -;&-21 m 5 m ( ¤ 14. ED &-21 PED f 3h& &-21 m 5 m ( HOH
43 ¢ F 7 G f -21 f 3h& -;&-21 m 5 m ( ¤ 15. &-21 ¢ 3hfI& ¢ 3hfI&0 ¡ = '8>?7A@CB 16. ¢ 3hfI& &-21 ¢ 3hfI&0 ¡ = '8>?7A@CB 17. a 1 ¢ 3hfI -21 -21 ¢ 3hfI ¢ 3hfI& 43 ¢ f ¢ 3hfI& -;&-21 ¢ 3hfI& ¤ 18. a 1 ¢ 3hfI -21 ¢ 3hfI&0 4 f ¢ 3hfI& ¤ 1.1.7. Inverse trigonometric functions
1. fI&0 3A &-21 3 ¢ f ¢ 3hf9 -; j &-21 F m 7( m G= '*>?7R@CB 3
< 2. 0fI&
3 ¢ &-21 3 ¢ f ¢ 3hf9 -; j &-21 F m 7 ( m G = '*>?7R@CB 3. fI&0
43 ¢ £ 3 ¢ f9 a 1 ; ¢ f9 O -;&-21 &-21 F 7 7<5 m G = '*>?7R@CB 4. a 1 fI&0
43 ¢ £ f9 a 1 ¢ f9 O -;&-21 a 1 F 7 7<5 m G ¤ 5. fI&0 43 ¢ a 1 £ 3 ¢ f9 a 1 ; ¢ f9 O -;&-21 &-21 F 7 7<5 m G = '*>h7A@CB
6. a 1 fI&0 43 ¢ a 1 £ f9 a 1 ¢ f9 O -;&-21 a 1 F 7 7<5 m G ¤
7. f;/ 0 ( ! ']" 3 ¢ f 0Q3hf -; j &-21 ) 7( m m m (8 , = '*>?7R@CB
8. f;/ 0 ')(:7X! % fI 1 -; f $ ¢ -; j + 1 -;9lC-; .&-21 £ f ¢ = '*>?7R@CB
9. EDM &-21 m H 3 '#" 3 ¢ f -21 0Q3hf9 -; j &-21 ) ( m m m (8 , = '*>?7R@CB
10. D &-21 m H ( 7! ' 3 ¢ fI &-!&4 0 f -; j + 1 -;9lC-; .&-21 ) m ¢ , = '*>?7R@CB
11. k &-21 ¢ 3hf & a 1 M ¢ 3hf & -21 f;/ F 7 G f9 -21 f;/ ¤
12. k a 1 ¢ 3hf9 & &-21 k -21 0f;/ F 72G f9 ¢ 3hf9 O& -21 0f / ¤
13. k -21 ¢ 3hf9 & a 1 k &-21 ¢ 3hf9 & -21 f / 43V -; £ -;&-21 0f;/ ¤
5
14. k a 1 ¢ 3hf & a 1 k -21 ¢ 3hf & -21 f;/
f9 f / ¤ 15. a 1 ¢ 3hf9 & -21 -21 ¢ 3hf9 & &-21 f /
£ ) 3 m , ¢ 3hf9 & -;&-21 f;/ ¤ 16. k 1 ¢ 3hf9 & &-21 &-21 f;/ 0 ¡ = '8>?7A@CB 17. k &-21 ¢ 3hf9 & 1 ¢ 3hf9 O& &-21 0f / ¡ = '8>?7A@CB 18. k &-21 ¢ 3hf & &-21 f;/ ¡ = '8>?7A@CB 19. k 1 ¢ f9 & &-21 f;/ ¡ = '8>?7A@CB 20. k &-21 ¢ f9 & P f;/ ¡ = '8>?7A@CB 21. k -21 ¢ f9 & Pk &-21 f /
£ F3 7 G -;&-21 0f;/ ¤ 22. k a 1 ¢ f9 & Pk -21 f;/
£ ) 3 m , f / ¤ 23. k &-21 M ¢ f & &-21 f;/
F 7 G f -21 ¢ f & -;&-21 f;/ ¤ 24. k a 1 k -21 ¢ f9 & &-21 0f;/
F 7 G f9 ¢ f9 & -;&-21 f;/ ¤ 1.2. The Hurwitz Zeta Function ζ(ν, z)
1.2.1. Derivatives with respect to the argument
1. fI& 3PfI ; < fI& ¤ 2. EDM &-21 F m GIH f -;&-21 F < m G ¤ 1.2.2. Derivatives with respect to the parameter
1.
& M]! ¤
7 $ £) 3 £ F '*G3 7 $ ' " £! 3 £ 3 ( 7X! " ' ! " &-21 1
' + -21 . F 'PG 3 ( 7! " ] $ (:7X! %O' ! " Y &-21 1
' F £) ' G 7' " 43 £! ¢ = ' ! = @O!C@CB 3.
F 7G - a 1 3 ' ' £ 1A- 3 ¢ 3 £ ¢ = = @ #C7 !C@MB
4.
F 7
G - a 1 C7( " ' ! C7( " ' !0' 3 ']" ' ( 7X! ' ' '#" '#" + &-21 . F 7 G " ' (:7 3 £ ¢ = = @ C7 !C@CB
5.
F 7 G - a 1
C7 (* " ' ! ' 3 C7( " ' ! '%& ' £ ( 7! " '#" '#" + &-21 . F 7 G 3 7( " ' '#" 3 £ ¢ = = @ C7 O!C@CB
6.
F
G - a 1 C7( " ' !AC7<5n " ' ! ' C7 ( " ' ! ' & ' £ C7(* " ' ! '#" ' ( 7X! " "" 5_7X! ']" '#" ']" + &-21 . F 7 G 7 (* " ' 7( " ' 3 £ ¢ = = @& !C@MB
1.3. The Exponential Integral Ei (z)
1.3.1. Derivatives with respect to the argument
1. 3PfI& 3 ¢ &-21 3 ¢ -; - &-21 m ]! "$&% = '8>?7A@ B 2.
3 ¢ -; - -;&-21 fI& = '8>?7A@ B :
!$ "!$ # 3. EDM &-21 F3 m G9H 3Q 3 ¢ -21 -R &-21 m ]! "$&% = '8>?7A@ B 4.
3 ¢ 3 ¢ -21 -R -;&-21 F m G = '8>?7A@ B 5. 3fI&0 f 43fI& f &-21 ( 7X! " $&% m ]! "'& = '8>?7A@ B 6. EDM &-21 R F]3 m GIH 43fI -;&-21 R F]3 m G 43 ¢ f &-21 -; &-21 3 ¢ F m G = '*>h7A@ B
7. -21 - P 3PfI& 3 ¢ -;&-21 43fI& ¤ 8. EDM a 1 -R ED -21 R F3 m G9H H 43 ¢ 4 F]3 m G ¤ 9. V- 3PfI& f 3PfI& ¤ 10. ED -R PED &-21 R F]3 m GIHOH f -;&-21 F]3 m G ¤ 1.4. The Sine si (z) and Cosine ci (z) Integrals
1.4.1. Derivatives with respect to the argument
1. 0fI& ')(:7X! % -; -;&-21 3AfI&3 V- -;&-21 AfI& = '8>?7A@ B
2. EDM &-21 F m GIH 43 ¢ ' (87! % D R -;&-21 F3 m G 3 #- R -;&-21 F m G9H = '*>?7R@CB
3. 0fI& ')(:7X! % -; -;&-21 43 AfI& - -;&-21 AfI& = '8>?7A@ B
4. EDM &-21 F m G9H 43 ¢ ')(87! %O D R -;&-21 F3 m G #- R -;&-21 F m G9H = '*>?7R@CB
5. &<3 0& 3 ¢ FO63 ' G &3 F Q3 ' G &
7 ' 7 ' & 1 3 £) ¢ 3P ¤ !<
; 6. & 0& 43 ¢
F Q3 ' G 0& F Q3 ' G 0&3 7 '%& 1 3 £! 3 ¤
1.5. The Error Functions erf (z) and erfc (z)
1.5.1. Derivatives with respect to the argument
1. 0fI& 43 ¢ &-21 m ' #- W W &-21 0fI& = '8>?7A@ B 2. 0f / ' (87! % m 1 -; - W 1 -;&-21 0f9 & = '8>?7A@ B 3. &-21 0f;/ ( 7! ']" '#" - W &-21 0f;/ = '8>?7A@ B 4. D 7 f;/ H ( 7! ' -;&-21 F 7 f G = '8>?7A@ B 5. DM &-21 F m G9H 3 m ' -;&-21 V- W W &-21 F m G = '8>?7A@ B 6. ED F m GIH 3 "' ']" - W &-21 F m G = '8>?7A@ B 7. D &-21 F m GIH 7 ) 7 m , = '8>?7A@ B 8. EDM &-21 F m G9H 43 ¢ '6(*7X! % m -21 - W 1 -;&-21 ) m , = ' > 7A@ B 9. D W W fI& H 43 AfI W W AfI&3 3 ' & 8 3PfI W W -;&-21 / £ fI ¤ 10. D &-21 W W m H AfI -;&-21 W W F m G
3 3 '%& 8 f -;&-21 W + W . -;&-21 ) m , ¤ 11. D W 0f / H ( m ! ' F 7 G
W F 7 3 _ f9 G ¤
L
! # " ( 12. ED &-21 W F m GIH m ' F 7G -;&-21 W ) 7 3 _ m , ¤ 13. D &-21 V- W D W 0f;/ HOH F 7 G f -21 0f;/ ¤ 14. ED a 1 - W PED &-21 W F m GIHOH
F 7 G f9 -;&-21 F m G ¤ 15. D a 1 W k -21 f / H F 7 G 3Pf9 W 0f / ¤ 16. ED &-21 W ED &-21 F m GIHOH
F 7 G 43f #- W F m G ¤ 17. ED W W 0fI& H 3 ' & 8 3PfI W W -;&-21 / £ fI ¤ 18. ED &-21 W W F m GIH 3 '%& 8 f -;&-21 W + W . -;&-21 ) m , ¤ 19. D &-21 W 0f;/ H "' £ -21 W A - &-21 f / £ ¤ 20. ED -21 W F m GIH
3 ¢ "' £ -;&-21 W + . - &-21 ) f * , ¤ 21. D -21 V- W D &-21 W 0f;/ HOH
43V -; £ -;&-21 0f / ¤ 22. ED a 1 - W ED -21 W F m GIHOH
43V -; £ F m G ¤ 23. ED 1 W &-21 f;/ 0 H ¡ = '8>?7A@ B 24. D a 1 - W D -21 W 0f / HOH f9 f / ¤
P
:1 25. ED &-21 - W ED &-21 W F m G9H H
f -;&-21 F m G ¤ 1.6. The Fresnel Integrals S(z) and C(z)
1.6.1. Derivatives with respect to the argument
1. fI&0 " m ')(:7! % 1 -; D 1 -;&-21 43 AfI&<3 - 1 -;&-21 AfI& H = '*>h7A@ B
2. EDM &-21 gF m GIH 43 ¢ " m ')(:7X! % -!&4 D R 1 -;&-21 F]3 m G 3 #- R 1 -;&-21 F m GIH= '*>?7R@CB
3. 0fI& " m ')(:7! % 1 -; D 1 -;&-21 43 AfI& #- 1 -;&-21 AfI& H = '*>h7A@ B
4. DM &-21 F m G9H 43 ¢ " m ')(:7X! % -!&4 D R 1 -;&-21 F]3 m G - R 1 -;&-21 F m GIH= '*>?7R@CB
1.7. The Generalized Fresnel Integrals S(z, ν) and C(z, ν)
1.7.1. Derivatives with respect to the argument
1. fI ;0 3 ')(:7X! % f# -;
-;&-21 3AfI&3 - -;&-21 AfI& = '*>?7R@CB
2. D &-21 F m GIH 43 ¢ &-21 ')(:7! % fV - -21 D R -;&-21 F3 m G 3 - R -;&-21 F m G9H= '*>?7R@CB
3. 0fI 3 ')(:7X! % f# -;
-;&-21 3AfI& - -;&-21 AfI& = '*>?7R@CB
,9N
4. EDM &-21 F m G9H
43 ¢ &-21 ')(:7! % fV - -21 D R -;&-21 F 3 m G 3 - R -;&-21 F m G9H= '*>?7R@CB 1.8. The Incomplete Gamma Functions γ(ν, z)
and Γ(ν, z)
1.8.1. Derivatives with respect to the argument
1. fI&0 3 ¢ fV -; - -;&-21 fI& = '8>?7A@ B 2. EDM &-21 F m GIH 3 ¢ 3 ¢ f# - -21 -R -;&-21 F m G = '8>?7A@ B 3. - fI&0 43 ¢ - -; < fI& ¤ 4. EDM a -21 F m G9H -21 F < m G ¤ 5. fI&0 ¢ 3 3PfI )3 < fI& ¤ 6. DM &-21 R F m G9H ¢ 3 f -;&-21 R F T3 _ m G ¤ 7. &- fI&0 ' %
f 1 N 1 J '65_7 ! m 5_7 L ¤ 8. D -21 R F m G9H 43 ¢ ' %
,
L1 , 18. ED &-21 F m G9H 43 ¢ &-21 3 ¢ fV - -21 -R -;&-21 0fI& = '8>?7A@ B 19. - fI& 3 ¢ - -; - _ fI& ¤ 20. ED a -21 F m GIH -21 F < m G ¤ 21. fI&0 ¢ 3 ; 43fI T3 _ fI& ¤ 22. D &-21 R F m GIH ¢ 3 ; f -;&-21 R F T3 < m G ¤ 23. &- fI& ¢ 3 ; f J '65_7 ! m 5_7 L ¤ 24. D -21 R F m GIH 3 ¢ ¢ 3 ; f -;&-21 ^ '65_7 ! 5_7 ` ¤ 25. ¢ 3 _ fI& 3 ¢ " m 1 -; - A &-21 F m G ¤ 26. ED &-21 F ¢ 3 < m GIH " m -!&4 -R + . &-21 F mOG ¤ 1.8.2. Derivatives with respect to the parameter
1.
1.9.1. Derivatives with respect to the argument
1. fI&0 F]3 m G J ' $ L £ 43 ; &- F m :G - 0fI& ¤ 2.
, ,
8. EDM a + -21 . - W A 0f;/ H 43 £ -; I+ -21 . - W A a 0f;/ ¤ 9. D W + . F m G9H
3 £ -; 43 -; a W + . - F m G ¤ 10. ED - + a 1 . - W + . F m G9H
£ -; -;&- + a 1 . - W + . a F m G ¤ 11. D -21 V- W A - f;/ H 3 £ -; -;&-21 V- W A ¤ 12. ED &-21 - W + . - F m G9H £ -; -21 - W + . ¤ 13. ED -21 - W A - &-21 f;/ H 3 ¢ £ -;&-21 O/ -;&-21 F f " G ¤ 14. ED - W + . - &-21 F m G9H £ -;&-21 O/ F m G ¤ 15. ED W A 0f / H
43 ¢ W A &-21 '65 $ ! ')( $ ! $9% ! '%&" f &- 3 ; &- -; a 0f;/ = '*>?7R@CB
16. ED V- W A f;/ H 43 ¢ - W A &-21 '65 $ !
')( $ ! $&% ! '%&" f &- a &- f;/ = '*>?7R@CB 17. D a 1 V- W A D -21 W A 0f;/ H H
F 7 ( G ) m , - W A 0f / ¤ 18. D &-21 - W + . P D &-21 W + . F m GIHOH
F 7 ( G ) m , -;&-21 - W + . F m G ¤ 19. D &-21 - W A D W A 0f / HOH
F]3 G ) m , -21 - W A f;/ ¤ ,93
P1 , 20. ED a 1 - W + . PED &-21 W + . F m GIHOH
F]3 G ) m , -;&-21 - W + . F m G ¤ 21. D &-21 W A D - W A 0f / HOH
F 5_7 G ) 3 m , -21 W A f;/ ¤ 22. D a 1 W + . D &-21 V- W + . F m GIHOH
F 5_7 G ) 3 m , -;&-21 W + . F m G ¤ 23. ED a 1 W A ED -21 #- W A 0f;/ H H
F ¢ G ) 3 m , W A 0f / ¤
24. ED &-21 W + . ED &-21 - W + . F m GIHOH F ¢ G ) 3 m , -;&-21 W + . F m G ¤
25. ED a 1 - W A ED &- -21 W A f;/ H H 3# -; 3 I+ -21 . -; - W A f;/ ¤
26. D &- -21 - W + . D W + . F m GIHOH 43V -; 3 ; - + a 1 . - W + . F m G ¤
27. D - -21 W A D a + -21 . - W A f / HOH 43# -; ¢ -;&- -21 W A f / ¤
28. ED a a 1 W + . EDM - + a 1 . V- W + . F m GIHOH 43# -; ¢ W + . F m G ¤
1.9.2. Derivatives with respect to the order
1.
M]! £ -;&-21 -
W F G Y 3 C 3 £ F 7 3 G F G 3h N
7 #7 ! W & #$
3 43 ¢ £ &-21 - W 1
"
#$
"
M]! a 1 £ -;&-!&4 - W a 1 F G Y
#$
3 ¢ £ -;&- £ ¢ -21 - W 1 ( " $ (:7! % 'T( $ 5_7X! % F 7 3 G
3 ¢ £ &-!&4 - W 1 7$
-21 &- ) , £ ¢ -!&4 &- a 1 ) , Y "" "" J 1 L)" 1 N 1
$ ! W $ 5 1 #$ 3 /
W 1 - -21 ) 3 , 3 43 ¢ £ &-21 / W -;&-21 ) 3 ,
3 ¢ ' % J & L ' a 1#-
W 1 N 1 '65_7 ! W 'g5 &
#$ 3 ' & V- W F]3 3 7 G ¤ 3.
& & -1 & N & 7 #7 ! 1
&
#$
£ &4 C £ £ -21 ) , 1 ) , 3
£ C £ 1 ) , 83 £ C 3 £ -21 ) , = @CB 1.10. The Bessel Function Jν(z)
1.10.1. Derivatives with respect to the argument
1. fI&0 F m G 3 ¢ J ' $ L 2 0fI& ¤ , ;
+ ( N1 2.
43& -; ( m ]! "')( $ ! % &- m ! "
% $ ( ! % a - 0fI& ¤ 3. EDM &-21 F m GIH F m G -;&-21 43 ¢ J ' $L 2 F m G ¤ 4. f;/ 0 F]3 7 G ¢ J ' $ L F G &- F m G f;/ ¤ 5. f / 0 F m G I+ -; . 2 f / ¤ 6. I+ a 1 . a 1 0f / 0 7 F m G &-21 f / ¤ 7. I+ a 1 . -;&-21 0f / ( 7! ' F m G &-21 0f / ¤ 8. - + a & . a 1 0f;/
/ *F m G a 1 a 1 F m G -;&-21 F m G ¤ 9. f / 0
F m G + -!&A . &-21 fI - '65 $ !$&% ')( $ ! - 2 0f / = '*>?7R@CB 10. ED &-21 F m GIH 7 ¢ J ' $ LF G &- F m G F m G ¤ 11. ED -21 F m GIH F m G - + . -21 F m G ¤ 12. EDI+ &- . a 1 F m GIH ( 7X! ' F m G &-21 -;&-21 F m G ¤ 13. EDI+ &- . -;&-21 F m GIH 7 F m G &-21 -;&-21 F m G ¤ 14. EDI+ &-21 . a 1 F m GIH
43 ¢ / 8F m G a 1 -21 a 1 F m G -;&-21 F m G ¤ 15. &-21 &-21 fI&0 m ! '#" &-21 = '8>?7A@ B
,9:
16. D -21 R &-21 F m G9H ( 7! ' £ fI &-21 - R = '8>?7A@ B 17. &-21 fI& &-21 fI&0 m ! '#" &-21 £ fI& ¤ 18. &-21 0fI& &-21 fI&0 m ! ']" &-21 £ fI& = '8>?7A@ B 19. &-21 fI& 1 -; fI&0 3 ¢ a 1 m ! '#" &-21 £ fI& = '*>?7A@ B 20. &-21 0fI& 1 -; fI&0 3 ¢ m ! '#" &-21 £ fI& ¤ 21. D -21 m &-21 F m GIH ( 7X! ' £ fI &-21 - m ¤ 22. ED -21 m &-21 F m GIH ( 7X! ' £ fI &-21 - m = '8>?7A@ B 23. 0f;/ F m G 43 ¢ J ' $ L a f;/ -; a 0f;/ ¤ 24. &-21 &-21 0f;/ m ']" I+ &-!& . &-21 £ f;/ = '8>?7A@ B 25. &-21 1 -; 0f / 3 m ']" I+ &-!& . &-21 £ f / = '8>?7A@ B 26. &-21 &-21 0f;/ 1 -; 0f;/
m '#" I+ &-!& . 1 -; £ f;/ = '*>?7R@CB 27. &-21 0f / a 1 0f / m '#" I+ &-21 . a 1 £ f / ¤ 28. D -21 &-21 F m GIH ( 7X! ' f &-21 - + a 1 . &-21 F m G = '*>?7R@CB 29. D -21 1 -; F m GIH ( 7X! '%& f &-21 - + a 1 . &-21 F m G= '*>?7R@CB 30. D -21 &-21 F m G 1 -; F m G9H
( 7X! ' f &-21 - + a 1 . 1 -; F m G ¤ , <
+ ( N1 , 31. ED -21 &-21 F m G a 1 F m G9H
( 7! ' f &-21 - + a & . a 1 F m G ¤ 32. ED &-21 F m GIH
F3 m G -; -21 43 ¢ J ' $L a F m G -; a F m G ¤ 1.10.2. Derivatives with respect to the order
1.
2.
M]! -21 *
£ & £ &0 = = @ B VC7 !C@MB 4.
M]! a 1 £ & a 1 0&3 43 ¢ £ & -;&-21 & ' % F G &-21 J L "$&% ' ( $ ! a 1 0&3 ' % 1 J L
"')( $ ! % $ Y -21
% &- a 1 & -21 £ &<3 3 ¢ &- - -;&-21 0& 1 - £ &0 ¤ 5.
M]! 1 -; £ & 1 -; &<3 3 ¢ £ & &-21 &
3 ' % &-21 J ( L ""'$&% ')( $ ! 1 - &<3 ' % 1 "')( $ ! % $
Y -21 ")"'&
% 3 ¢ -; a 1 0& -21 £ &3 3 ¢ a &- -21 & 1 - £ &0 ¤ 6.
M]! 1 ¢ &-21 £ a 1 / -;&-21
Y &-21 ')( $ ! % J L "
$9% ')(* $ ! % J $ 5 1 L J $ (*'65 1 L DXF F 7G 3 F 3 7GG ,9L
Y
£ & H ¢ £ &-21 / -; a 1 &-21 ')( $ (:7X! % J L "
$&% ')(* $ (:7! % J $ 5 & L J $ (*'65 1 LY DXF F G 3 F 3 7GG 3 £ &
£ & H = '*>h7A@CB 1.11. The Bessel Function Yν(z)
1.11.1. Derivatives with respect to the argument
1. 0fI& F m G 3 ¢ J ' $ L 2 0fI& ¤ 2.
43& -; ( m ]! "')( $ ! % &- m ! "
% $ ( ! % a - 0fI& ¤ 3. D &-21 F m GIH F m G -;&-21 43 ¢ J ' $ L 2 F m G ¤ 4. f / 0 F m G I+ -; . 2 0f / ¤ 5. I+ a 1 . a 1 0f;/ 3 7 F m G &-21 f;/ ¤ 6. I+ a 1 . -;&-21 0f / ( 7X! ' F m G &-21 f / ¤ 7. - + a & . -;&-21 0f /
43 ¢ a 1 / 8F mOG a 1 a 1 F m G -;&-21 F m G ¤ 8. f / 0 F m G I+ -!&A . &-21 fI - '65 $ !$9% ')( $ ! - 2 f / = '*>h7A@CB
9. D -21 F m G9H F m G I+ -; . -21 F m G ¤ 10. EDI+ &- . a 1 F m GIH ( 7X! '%& F m G &-21 -;&-21 F m G ¤
,9P
+ ( 11. EDI+ &- . -;&-21 F m G9H 7 F m G &-21 -;&-21 F m G ¤ 12. D + &-21 . -;&-21 F m G9H
3 / *F mG a 1 -21 a 1 F m G -;&-21 F m G ¤ 13. &-21 1 -; fI&0 3 ¢ a 1 m ! '#" &-21 = '8>?7A@ B 14. D -21 R 1 -; F m GIH 3 m ! '#" - R = '8>?7A@ B 15. &-21 1 -; 0f;/ 3 ¢ a 1 m ']" I+ &-!& . 1 -; £ f;/ = '*>?7R@CB 16. &-21 &-21 0f;/ 1 -; 0f;/
3 ¢ a 1 m ']" I+ &-!& . &-21 £ f;/ = '8>?7A@ B 17. &-21 0f;/ a 1 0f;/
43 ¢ a 1 m '#" I+ &-21 . -;&-21 £ f;/ ¤ 18. ED -21 1 -; F m GIH 3 m '#" - + a 1 . 1 -; F m G = '8>?7A@ B 19. D -21 &-21 F m G 1 -; F m G9H
3 m ']" - + a 1 . &-21 F m G ¤ 20. ED -21 &-21 F m G a 1 F m G9H
3 7 f &-21 - + a & . -;&-21 F m G ¤ 21. &-21 &-21 0f / 1 -; 0f /
3 ¢ a 1 m ']" I+ &-!& . &-21 £ f / = '8>?7A@ B 22. &-21 1 -; 0f;/ &-21 0f;/
3 ¢ a 1 m ']" I+ &-!& . &-21 £ f;/ = '8>?7A@ B 3 N
, 3 8 , 3 8
23. &-21 &-21 0f;/ &-21 0f;/ ( 7X! ' f &-21 I+ &-!& . 1 -; £ f / ¤
24. &-21 0f / -;&-21 0f / ( 7X! ' f &-21 I+ &-21 . a 1 £ f;/ ¤
25. a 1 0f;/ 1 -; 0f;/ ( 7X! '%& f &-21 I+ &-21 . a 1 £ f;/ ¤
26. D -21 &-21 F m G &-21 F m G9H m ']" - + a 1 . 1 -; F m G ¤
27. ED -21 1 -; F m G &-21 F m G9H 3 m ']" - + a 1 . &-21 F m G ¤
28. D -21 &-21 F m G -;&-21 F m GIH m '#" - + a & . a 1 F m G ¤ 29. ED -21 a 1 F m G 1 -; F m G9H
3 m ']" f &-21 - + a & . a 1 F m G ¤ 1.11.2. Derivatives with respect to the order
1.
2.
ν (z) and H(2) ν (z)
1.12.1. Derivatives with respect to the argument
1. + . f / 0 F m G I+ -; . + . 2 0f / = 7]X@CB
2. EDM -21 + . F m G9H F m G I+ -; . -21 + . F m G = 7]X@CB
31
+ ( !, , 3. I+ a 1 . + . a 1 0f / ( 7X! F m G &-21 + -21 . U = 7]X@CB 4. DM &-21 + 1 .1 -; 0f;/ + .1 -; f;/ H ¡ = '8>?7A@CB 1.12.2. Derivatives with respect to the order
1.
3 8 M]! 3 ¢ + . & ' % &-21 J L ""'$&% ')( $ ! + . 0& = 7]X@CB
2.
3 8 M]! -; 3 ¢ a + . &<3 3 ¢ ' % &-21 J L
""'$9% ')( $ ! + . &= 7@CB 3.
3 8 M]! 1 *
Y + -21 . 3 ¢ a 1 £ &<3 £ & 43 ¢ = 7@CB 4.
3 8 M]! -21 *
Y + -21 . £ & 3 ¢ a 1 £ &0 3 ¢ = 7@CB 1.13. The Modified Bessel Function Iν(z)
1.13.1. Derivatives with respect to the argument
1. fI&0 F m G J ' $L 2 0fI& ¤ 2.
43& -; ( m ]! "')( $ ! % &- m ]! " % $ ( ! % - a 0fI& ¤
3. DM &-21 F m GIH F3 m G -;&-21 J ' $ L 2 F m G ¤ 4. f;/ 0 F]3 7G J ' $L F G &- F]3 m G f;/ ¤ 5. f;/ 0 F m G I+ -; . 2 0f;/ ¤ 6. I+ a 1 . a 1 f / 0 7 F m G &-21 0f / ¤
3,
7. I+ a 1 . -;&-21 f / 0 7 F m G &-21 0f / ¤ 8. - + a & . a 1 f;/ 0
/ 8F mO G a 1 a 1 F m G -;&-21 F m G ¤ 9. k 0f /
F m G I+ -!&A . &-21 3PfI - '65 $ !$&% ' ( $ ! - 2 f / = '*>?7R@CB 10. D &-21 F m G9H 7 J ' $ L F G &- F3 m G F m G ¤ 11. ED -21 F m G9H F]3 m G - + . -21 F m G ¤ 12. EDI+ &- . a 1 F m GIH ( 7! ' F m G &-21 -;&-21 F m G ¤ 13. EDI+ &- . -;&-21 F m GIH ( 7X! ' F m G &-21 -;&-21 F m G ¤ 14. D I+ &-21 . a 1 F m G9H
3 ¢ / *F m G a 1 -21 a 1 F m G -;&-21 F m G ¤ 15. &-21 &-21 0fI& m ! '#" &-21 = '8>?7A@CB 16. k &-21 - a 1 fI& m ! "'#" -;&-21 £ ¢!!£ fI& ¤ 17. k &-21 fI& £ fI # -21 J 5n'65 1 L 5_7X! 1 N 1 5n'65 1 5_7 ! m ¤ 18. ED -21 R &-21 F m GIH ( 7X! ' £ fI &-21 - R = '8>?7A@CB 19. D -21 V-R a 1 m H 43 ¢ m ! "'#" F £ ¢) m G ¤ 20. ED 7 R F m GIH
#$ ¤
3 3
+ ( 31 21. &-21 0fI& &-21 0fI& m ! ']" &-21 £ fI& ¤ 22. &-21 0fI& &-21 0fI& m ! '#" &-21 £ fI& = '8>?7A@CB 23. ED -21 m &-21 F m GIH ( 7! ' £ fI &-21 - m ¤ 24. D -21 m &-21 F m G9H ( 7! ' £ fI &-21 - m = '8>?7A@CB 25. V f / F m G J ' $ L a 0f / -; a 0f / ¤ 26. &-21 V &-21 f;/ 0 m '#" I+ &-!& . &-21 £ f;/ = '8>?7A@CB 27. &-21 V 1 -; f / 0 m '#" I+ &-!& . &-21 £ f / = '8>?7A@CB 28. &-21 &-21 f;/ 1 -; f;/ 0
m ']" I+ &-!& . 1 -; £ f;/ = '*>?7R@CB 29. &-21 0f / a 1 f / 0 m ']" I+ &-21 . a 1 £ f / ¤ 30. ED &-21 F m GIH
F3 m G -; -21 KJ ' $L a F m G -; a F m G ¤ 31. D -21 V &-21 F m G9H ( 7! ' f &-21 - + a 1 . &-21 F m G = '*>?7R@CB 32. ED -21 1 -; F m G9H ( 7! ' f &-21 - + a 1 . &-21 F m G = '*>?7R@CB 33. ED -21 &-21 F m G 1 -; F m GIH
( 7X! ' f &-21 - + a 1 . 1 -; F m G = '8>?7A@CB 34. ED -21 &-21 F m G a 1 F m GIH
( 7! ' f &-21 - + a & . a 1 F m G ¤ 395
35. a - P - 0fI& F 7G £ fI - 0fI& ¤ 36. ED &- - R PED a &-21 R F m GIHOH
F 7 G £ fI - -;&-21 -R F m G ¤ 37. &- P - 0fI& F 7 3 G 3 £ fI - fI& ¤ 38. ED a R ED &- -21 V-R F m GIHOH
F 7 3 G 3 £ fI -;&-21 R F m G ¤ 39. &- - < 0fI& F 7 3 G £ fI - - 0fI& ¤ 40. D a #- R D &- -21 R F m GIHOH
F 7 3 G £ fI -;&-21 -R F m G ¤ 41. a P - - 0fI& F 7 G 3 £ fI fI& ¤ 42. D &- R P D a -21 -R F m GIHOH
F - 7G 43 £ fI - -;&-21 R F m G ¤ 1.13.2. Derivatives with respect to the order
1.
M]! 1 * 7 3 £ & #- £ & = = @ B C7 !C@MB
3. *
M]! a 1
7 3 £ & -;&-21 & a 1 &03 ( 7X! ' £ & a 1 0& 643 ¢ ' % &-21 ( 7X! "$9% ')( $ ! F ,G -; a 1 &
3 ' % " 1 J (
L "')( $ ! % $ -21
( ! %Y 3 ¢ &- a 1 & -;&-21 0& -21 £ &3 43 ¢ &- &- a 1 & -21 £ & 1 - £ &0 ¤3;
+ ( 5) 5.
7 £ &3 £ &00 -;&-21 & a 1 &0&3 ( 7! ' a 1 &Y £ & £ & 43 ¢ ' % &-21 ( 7X! "$&% ')( $ ! F G -; a 1 0&
3 ' % " 1 J (
L "')( $ ! % $ -21
(]! %Y 3 ¢ &- a 1 & -;&-21 0& -21 £ &3 43 ¢ &- &- a 1 & -21 £ & 1 - £ &0 ¤
6.
M]! 1 -;
7 £ &<3 £ &00 -;&-21 & a 1 &03 ( 7! ' £ & £ & &-21 &
3 ' % &-21 J ( L ""'$9% ')( $ ! 1 - & ' % " 1 J L
"')( $ ! % $ -21
%Y 3 ¢ -21 &- -21 & -; a 1 0& -21 £ & 43 ¢ &- &- -21 & -21 £ & 1 - £ &0 ¤ 7.
M]! 1 £ &-21 O/ -;&-21 &-21 ( 7! " ')( $ ! % J L
"
$9% ')(* $ ! % J $ 5 1 L J $ (*'65 1 LY D £ F F 7 G 3 F 3 7 G;G 3 £ & V- £ & H
£ &-!&4 / -; a 1 &-21 ( 7! " ')( $ (:7! % J L "
$9% ')(* $ (:7X! % J $ 5 & L J $ (*'65 1 LY D £ F F G 3 F 3 7GG 3 43 £ & - £ & H ¤
1.14. The Macdonald Function Kν(z)
1.14.1. Derivatives with respect to the argument
1. fI&0 F]3 m G J ' $ L 2 0fI& ¤ 2.
43& -; m ]! "')( $ ! % &- ( m ! " % $ ( ! % - a 0fI& ¤
3 :
$ 3. EDM &-21 F m GIH F m G -;&-21 J ' $ L 2 F m G ¤ 4. f;/ F3 m G I+ -; . 2 0f;/ ¤ 5. I+ a 1 . a 1 f;/ 0 43 ¢ '%& f &-21 - ¤ 6. - + a & . a 1 f;/ 0 ( 7! ' F m G a 1 -;&-21 a 1 F m G ¤ 7. f /
F3 m G I+ -!&A . &-21 f - '65 $ !$&% ')( $ ! - 2 f / = '*>?7R@CB 8. EDM -21 F m GIH F m G - + . -21 F m G ¤ 9. DM + &-21 . a 1 F m G9H 7 F m G a 1 -21 a 1 F m G ¤ 10. k &-21 a 1 0fI& 3 ¢ £ / 6 £ fI -;&-21 -;&-21 ¤ 11. k SEa 1 )SEa 1 fI& 3 ¢ SEa _ / g £ fI &- S -21 &- S -21S -; £ fI& = > 'I@ B 12. k &-21 )SEa 1 fI& 43 ¢ S / 6 £ fI - S -21 - S -21 - S -21S -; £ fI& = > 'I@ B 13. k - S -21 )SEa 1 fI& 3 ¢ SEa / g £ fI - S -21 - S -;&-21 - S -;&-21S £ fI& ¤ 14. k SEa 1 - )SEa 1 fI& 3 ¢ SEa _ / g £ fI &- S -21 - &- S -21S £ fI& ¤ 15. k &-21 - )SEa 1 fI& 43 ¢ S / 6 £ fI - S -21 - S -21 - - S -21SEa £ fI& ¤ 16. k - S -21 #- )SEa 1 fI& 3 ¢ S / g £ fI - S -21 - S -;&-21 #- - S -;&-21SEa £ fI& ¤ 17. D -21 R )SEa 1 F m GIH
43 ¢ SEa / 6 £ fI - S -21 S -; - S -21S -; F m G = >_'I@CB 3<
+ ( 5) 18. ED SEa &-21 R )SEa 1 F m GIH
3 ¢ S / g £ fI - S -21 S - S -;&-21S F m8G ¤ 19. ED &- S -!&4 R )SEa 1 F m G9H
3 ¢ S ? / g £ fI &- S -21 -;&-21 &- S -21S -; F m G = > 'I@ B 20. D -21 V-R )SEa 1 F m GIH
3 ¢ SEa / g £ fI - S -21 S -; - R - S -21SEa F m G ¤ 21. ED SEa &-21 -R )SEa 1 F m G9H
3 ¢ SEa / g £ fI - S -21 S - R - S -;&-21SEa F m:G ¤ 22. ED &- S -!&4 -R )SEa 1 F m G9H
43 ¢ S _ / g £ fI &- S -21 -;&-21 V- R &- S -21S F m G ¤ 23. 0f;/ 0 F]3 m G J ' $ L a f;/ -; a 0f;/ ¤ 24. &-21 &-21 f;/ 0 43 ¢ / f &-21 + &-!& . &-21 £ f;/ ¤ 25. &-21 0f;/ a 1 f;/ 0
43 ¢ / f &-21 I+ &-21 . a 1 £ f;/ ¤ 26. ED &-21 F m G9H
F m G -; -21 J ' $ L a F m G -; a F m G ¤ 27. D -21 &-21 F m G9H / f &-21 - + a 1 . &-21 F m G ¤ 28. D -21 &-21 F m G a 1 F m GIH
/ f &-21 O - + a & . a 1 F m G ¤ 29. &-21 &-21 f / &-21 f / 0
m ']" I+ &-!& . &-21 £ f;/ = '*>?7R@CB 3 L
$ 30. ED -21 &-21 F m G &-21 F m GIH
( 7X! ' f &-21 - + a 1 . &-21 F m G = '8>?7A@CB 31. - &-21 fI&0 3 ¢ F 7 3 G F 7 G -;&-21 - 0fI& ¤ 32. ED - R PED -21 R F m GIHOH
3 ¢ F 7 3 G F 7 G -21 V-R F m G ¤
33. < &-21 - fI&0 43 ¢ F 7 3 G F 7 G -;&-21 0fI& ¤ 34. ED R PED -21 -R F m GIHOH
3 ¢ F 7 3 G F 7 G -21 R F m G ¤
35. &- S -21 - SEa 1 )SEa 1 fI&0 3 £ fI - S -21 #- )SEa 1 fI& ¤ 36. ED SEa a 1 - R PED &- S -!&4 R )SEa 1 F m G9HOH 43 £ fI S -;&-21 -R )SEa 1 F m G ¤ 37. a SEa 1 < - S -21 - )SEa 1 fI&0 5n'! % % 43 £ fI SEa 1 )SEa 1 fI& ¤ 38. ED &- S -21 R PED a2S -21 -R )SEa 1 F m G9HOH
5n'! % % 3 £ fI - S -;&-!&4 R )SEa 1 F m G ¤ 39. a SEa 1 - P - S -21 )SEa 1 fI&0 5h'! % % £ fI SEa 1 - )SEa 1 fI& ¤ 40. ED &- S -21 - R PED a2S -21 R )SEa 1 F m G9HOH
5n'! % % £ fI - S -;&-!&4 V-R )SEa 1 F m G ¤ 41. - &-21 )SEa 1 fI&0 5n'! % (*'! % -;&-21 - )SEa 1 fI& ¤
3 P
+ ( 5) , 42. ED - R PED -21 R )SEa 1 F m GIHOH
5n'! % (*'! % -21 -R )SEa 1 F m G ¤ 1.14.2. Derivatives with respect to the order
1.
M! ¡\¤
2.
M!
4.
' % &-21 J L ""'$&% ' ( $ ! -21 0&3 43 ¢ ' % / 1 J ( L
"')( $ ! % $ &- -21 0& -; a 1 0& -21
!& % -21 £ & ¤
6. 3 ¢ £ &<3 £ &00 &-21 & 1 -; 0& ' % &-21 J L ""'$9% ')( $ ! -21 &3 3 ¢ ' % / 1 J (
L "')( $ ! % $ &- -21 &
% -21 £ & ¤ 7.
"
"
$9% ')(* $ (:7X! % J $ 5 & L J $ (*'65 1 LY D - F F G 3 F 3 7 GG 3 3 £ & H ¤ 5N
H
L
1. H fI&0 F3 G J ( 1 L
"$&% ')(* $ ! % &- J ')( $
L<F G &- - F]3 m nG H - 0fI& ¤ 2.
F]3 G J ( 1 L "$9% ')(* $ ! % &-
J ' ( $ L F]3 G &- - Y F m nG H a 0fI&3 7
F m nG a -21 -21
J 5 1 L
¤
3. H f;/ 0 F m G I+ -; . H -; f;/ ¤ 4. - H f / 0 F3 m G - + a . H a 0f /
3 ( 7X! ' F m G a &-21 -21 &-21 J $ 5 1 L J 5n')( $ 5 1 L F
m G ¤ 5. + a 1 . H a 1 f;/ 0 F m G &-21 m ¤
6. EDM &- -21 H F m G9H F]3 m G - + a . -21 H -; F m G ¤ 7. D I+ &- . H a 1 F m G9H 43 ¢ F m G &-21 -;&-21 m ¤ 8. DM a -21 H F m G9H F m G + -; . -21 H a F m G
3 7 F m G a &-21 -;&-21 &-21 J $ 5 1 L
J 5:' ( $ 5 1 L F m G ¤ 9. L 0fI&
F]3 G J ( 1 L "$&% ')(* $ ! % &-
J ')( $ L F G &- - F]3 m nG L - 0fI& ¤
5)
+ ( !; , 10.
F 3 G J ( 1 L "$&% ')(* $ ! % &-
43 ¢ J ')( $ L F 3 G &- - Y F m G L a fI& 7
F m G a -21 -21
J 5 1 L J 5 Q(
5 1 L F]3
m G
¤
11. L 0f / F m G I+ -; . L -; f / ¤ 12. - L 0f;/ F m G - + a . L a f;/
7 F m G a &-21 -1 &-21 J $ 5 1 L
J 5n')( $ 5 1 L F 3 m G ¤
13. I+ a 1 . L a 1 0f;/ F m G &-21 m ¤
14. ED &- -21 L F m GIH F3 m G - + a . -21 L -; F m G ¤ 15. EDI+ &- . L a 1 F m GIH 3 ¢ F m G &-21 -I&-21 m ¤ 16. ED a -21 L F m GIH F3 m G I+ -; . -21 L a F m G
( 7X! ' F m G a &-21 -;&-21 &-21 J $ 5 1 L
J 5n')( $ 5 1 L F3 m G ¤ 1.15.2. Derivatives with respect to the order
1.
H M! 3 & ']" "'
&
<!¡
7 &-21 J 1 L)"J 1 L ']")" F G &- -21 D 3 F 3 7 GIH
' % &-21 3 ¢ J L ""'$&% ')( $ ! H - & = Q> @CB 5,
H
L
3.
&
!_!¡
H M! 1 *
Y C £ &3 £ &0 £ &3 £ 0& = = @ B #O !C@CB 5.
H M! -1 *
Y £ &3 £ &0&3 £ &<3 £ &0 = = @ B # !C@CB 6.
H M]! a 1 £ &3 £ &0 a 1 0& 3 ¢ £ &3 £ 0& -I&-21 0& H a 1 &<3 a 1 &0 7 F G a 1 F 7G D C £ £ F 7 3 GIH
3 ' % F G &-21 J ( L "$&% ' ( $ ! - -21 0&3 ' % F G a 1 &-21 J 1 L)"$9% ')( $ ! 3 J L ']"
&-21 J 1 L"')( $ ! % F G 3 ¢ 3 / 8F G 1 -; &-21 J L "$&% ' ( $ ! &- -21
J L %Y 43 ¢ a 1 a 1 0& 1 - &<3 £ -21 1 - £ & 3?43 ¢ - -21 & -21 0&3 £ -21 -21 £ & ¤ 7.
L M! &<3 7
¡
3 7 &-21 43 ¢ J 1 L "J 1 L'#")" F G &- -21 D 3 F 3 7 G9H
' % &-21 J ( L ""'$9% ')( $ ! L - 0& = Q> @CB 12.
L M]! -; 3 ¢ & ( ! '#" "'
L M]! 1 *
3 7 7 F C ,G - £ &<3 £ 0& 3 43 £ &<3 £ 3P& = = @ B C7 !C@MB 14.
L M]! 1 *
15.
£ &<3 £ &03 £ &<3 £ &0 ¤ 16.
L M]! a 1 £ &3 £ &0 a 1 0&3 £ &<3 £ &0 -;&-21 0& L a 1 &<3 -;&-21 0& ( 7X! '%& F G a 1 F 7 G D C £ £ F 7 3 GIH
' % F3 G &-21 J ( L "$9% ')( $ ! - -21 & 43 ¢ ' % F G a 1 &-21 J 1 L "$9% ')( $ !
5 5
J L '#" &-21 3 ¢ J 1 L "')( $ ! % F G 3 ¢
43 ¢ / 8F ,G 1 -; &-21 J ( L "$&% ' ( $ ! &- -21 J ( L %Y a 1 & 1 - &<3 £ -21 1 - £ & 3 - -21 0& -21 &<3 £ -21 -21 £ & ¤
17.
L M]! -I&-21 £ &<3 £ & a 1 0& £ &<3 £ &0 -;&-21 & 3 ' % &-21 J ( L
""'$9% ')( $ ! a 1 & ' % / 1 J ( L "')( $ ! % $ -21
( ! %Y &- a 1 & £ 1 - -21 &<3 -21 £ & 3 -;&-21 0& £ 1 - 1 - &<3 1 - £ & ¤
1.16. The Anger Jν(z) and Weber Eν(z) Functions
1.16.1. Derivatives with respect to the argument
1. J 0& F3 ,G -; J ( 1 L "$9% ')(* $ ! % &-
J ')( $ L F G \ \ F]3 ,G
Y J - &<3 3 ¢ ! -21
F Q( ( 5 7 G
F G ¤
"$&% ')(* $ ! % &- J ')( $
5 5_7 G
F]3 G ¤
3. F 7G 43 ¢ J ' $,L J 2 & ¤
4. J 0f;/ F m G + -; . J -; 0f;/ 3 F3 m G ! m I+ -;&-21 . &-21 F ' ( $ ( 5 7 G F m G ¤
5;
+ ( :1 5. - J 0f;/ F3 m G - + a . J a f;/
3 F m G ! m - + a a 1 . &-21 F ')( $ 5 5_7 G F]3 m G ¤ 6. EDM &- -21 J F m GIH F3 m G - + a . -21 J -; F m G
3 F m G ! m - + a a 1 . &-21 F ')( $ ( 5_7 G F m G ¤ 7. D a -21 J F m GIH F m G I+ -; . -21 J a F m G
3 F]3 m G ! m I+ -;&-21 . &-21 F ')( $ 5 5_7 G F]3 m G ¤ 8. E &0 F 3 G -; J ( 1 L
"$9% ')( $ ! % &- J ')( $
E - & 7 -21
F G ¤
"$&% ')(* $ ! % &- J ')( $
E a & 7 -21
F G ¤
10. F 7G 43 ¢ J ' $ L E 2 & ¤
11. E f;/ 0 F m G + -; . E -; 0f;/ 7 m F m G + -;&-21 . Y &-21 3 ¢ 43 ¢ XF ')( $ ( 5_7 G F m G ¤
12. - E f;/ 0 F]3 m G - + a . E a 0f;/ 7
m F m G - + a a 1 . Y &-21 3 ¢ 3 ¢ XF ')( $ 5 5_7 G F m G ¤ 5:
Jν(z)
Eν(z)
13. D &- -21 E F m G9H F]3 m G - + a . -21 E -; F m G F3 m G " 3 '&(' & 8
mY &-21 3 ¢ 3 ¢ XF ')( $ ( 5_7 G F m G ¤ 14. ED a -21 E F m G9H
F m G I+ -; . -21 E a F m G F 3 m G 3 "']" 8 mY &-21 3 ¢ 43 ¢ XF ')( $ 5 5_7 G F m G ¤
1.16.2. Derivatives with respect to the order
1.
' % &-21 J L '#")"$9% ')( $ ! 0& H &
3 7 &-21 J 1 L "J 1 L '#")" F ,G &- -21 ( 7X! ' &-21 3 ¢ F ')( $ 5_7 G F G ¤ 2.
J M! -I 43 ¢ &-21 ' % &-21 J L
']")"$&% ')( $ ! & H -; & ( 7X! ' &-21 F ')( $ 5_7 G F G ¤
3.
-21 J 1 L J 1 L)""
F ,G - -21 7 &-21 3 ¢ 3 ¢ XF ')( $ 5_7 G F]3 G a 1 -21
7 5n')( $ 5 7 ¤
' % &-21 J ( L ']")"$&% ' ( $ ! H - & 3 ¢ &
7 &-21 43 ¢ 43 ¢ XF ')( $ 5_7 G F G a 1 -21 7
5n')( $ 5 7 ¤
and keiν(z)
1.17.1. Derivatives with respect to the argument
1. f;/ 0 F m G I+ -; . D ' 2 0f;/ <3
' 2 0f;/ H ¤ 2. f;/ 0
F m G + -; . D ' 2 0f;/ ' 2 0f;/ H ¤
3. I+ a 1 . a 1 f / 0 3 7 F m G &-21 Y D O')(:7X! FOf " G FOf " G ')(:7! F f " PG F f " G9H ¤
4. I+ a 1 . a 1 0f;/ 7 F m G &-21 Y D O')(87! FOf " PG F f " PG 3
')(:7! F f " G F f " G9H ¤ 5. I+ a 1 . -;&-21 0f / ( 7! ' & F m G &-21 Y D O')(87! FOf " G F f " G3
')(:7X! F f " G F f " G9H ¤ 6. I+ a 1 . -;&-21 0f;/ ( 7X! ' F m G &-21 Y D ')(:7X! F f " G FOf " PG
O')(:7X! F f " G F f " G9H ¤ 7. k - + a & . a 1 0f / / F mO G a 1 Y O'65_7X! D a 1 F m G -;&-21 F m G
3 a 1 F m G -;&-21 F m G9H 5L
# " "! #$" #$$! 3
O'65 7! D a 1 F m G -;&-21 F m G a 1 F m G -I&-21 F m GIH ¤ 8. k - + a & . a 1 f;/ / 8F mOG a 1 Y '65_7! D a 1 F m G -I&-21 F m G3 a 1 F m G -;&-21 F m G9H
'65_7! D a 1 F m G -;&-21 F m G a 1 F m G -I&-21 F m GIH ¤ 9. EDM &-21 A F m &-21 0fI& m &-21 fI& G9H
m ! '#" &-21 D O')(87! / £ fI H = '8>?7A@ B 10. ED &-21 A F m &-21 0fI&3 m &-21 fI& G9H
m ! '#" &-21 D O')(87! / £ fI H = '8>?7A@ B 11. D &-21 F m m &-21 fI&3 m m &-21 fI& G9H
m ! '#" &-21 D ' (87! / £ fI / £ fI 3
O')(87! / £ fI / £ fI H ¤ 12. D &-21 F m m &-21 fI& m m &-21 fI& G9H
m ! ']" &-21 D O')(:7X! / £ fI / £ fI O')(87! / £ fI / £ fI H ¤
13. k &-21 <1 f / <3 <1 f / m ']" I+ &-!& . Y D O')(:7X! &-21 £ f;/ <3
O')(:7X! &-21 £ f;/ H= '*>?7R@CB 5P
+ ( !< 14. k &-21 <1 0f / <1 f /
m ']" I+ &-!& . Y D O')(87! &-21 £ f / O')(:7X! &-21 £ f / H= '*>?7R@CB
15. 0f;/ F m G I+ -; . 2D ' 2 0f / 3
' 2 0f / H ¤ 16. 0f;/
F m G I+ -; . 2D ' 2 f;/ ' 2 0f;/ H ¤
17. + a 1 . a 1 0f;/ 43 ¢ ' & f &-21 - / A Df " O'65 ! H ¤
18. I+ a 1 . a 1 0f;/ 43 ¢ a 1 '%& f &-21 - / A D f " O'65 ! H ¤
19. k - + a & . a 1 f;/ ( 7! ' F mO G a 1 Y O'65_7X! D a 1 F m G 3 a 1 F m G9H 3 £ '65_7X! a 1 F m G a 1 F m G ¤
20. k - + a & . a 1 f;/ ( 7X! ' F m G a 1 Y '65_7! D a 1 F m G 3 a 1 F m GIH £ '65_7X! a 1 F m G a 1 F m G ¤ 21. k &-21 &-21 f / &-21 0f / 3 ¢ f &-21 I+ &-!& . Y D ')(:7! &-21 £ f;/ <3
O')(:7X! &-21 £ f;/ H ¤ 22. k &-21 &-21 0f;/ 3 &-21 0f;/
43 ¢ / f &-21 I+ &-!& . Y D ')(:7! &-21 £ f;/ O')(:7X! &-21 £ f;/ H ¤
;9N
# " "! #$" #$$! 23. k &-21 &-21 0f;/ &-21 f;/ 3 &-21 0f;/ &-21 f;/ 4
m '#" I+ &-!& . D O')(:7X! &-21 £ f / 3
O')(87! &-21 £ f;/ H ¤ 24. k &-21 &-21 0f;/ &-21 f;/ &-21 0f;/ &-21 0f;/ 4
m ']" + &-!& . D ')(:7! &-21 £ f;/ O')(:7X! &-21 £ f / H ¤
1.17.2. Derivatives with respect to the order
1.
M! 3 0&3 & ' % &-21 J L ""'$9% ')( $ ! D ] $ (*'! 0& # $ (*'! & H ¤
2.
# $ (*'! & H ¤ 3.
$ (*'! & H ¤ 6.
;
7#7 ! ( 1
1 & & & #$
1 1 & #$
& ! ( 1
')( $ ! a 1 & H 3 ' % 1 J L
"')( $ ! % $ -21
% &- a 1 0&Y D $ (* (:7X! -21 £ & $ (*Q(:7X! -21 £ & H &- a 1 &&D $ (* (:7! -21 £ &3
$ (* (:7! -21 £ & H3 43 ¢ a a -;&-21 0&D $ (*Q(87! 1 - £ & $ (*Q(:7X! 1 - £ & H3 3 ¢ a a -;&-21 &&D $ (*Q(:7X! 1 - £ &3
$ (* (:7! 1 - £ & H = @ B 8.
M! a 1 M! a 1
7 a 1 & # 1 N 1 ! (
1 & & & #$
1 !
& ! ( 1
')( $ ! a 1 & H ' % 1 J L
"')( $ ! % $ -21
% &- a 1 0&Y D $ ( Q(87! -21 £ &<3 $ ( (:7! -21 £ & H3 &- a 1 0&D $ (*Q(:7X! -21 £ &
$ (* (:7! -21 £ & H3 3 ¢ a a -;&-21 &&D $ (*Q(:7X! 1 - £ &3 $ (*Q(:7X! 1 - £ & H 43 ¢ a a -;&-21 &&D $ (*Q(87! 1 - £ &
$ (* (:7X! 1 - £ & H = @ @ B 9.
M! 1 -; 7
1 & & & #$
1 1 & #$
& ! ( 1
3 ' % &-21 J ( L ""'$9% ')( $ ! D ')( $ ! 1 - & ')( $ ! 1 - 0& H
;93
"')( $ ! % $ -21
% 3 ¢ a &- -21 0&Y D $ (*Q(:7X! 1 - £ & $ ( (:7! 1 - £ & H 3 ¢ a &- -21 &&D $ (* (:7! 1 - £ &3
$ (*Q(:7X! 1 - £ & H3 43 ¢ -; a 1 &&D $ (*Q(87! -21 £ & $ (* (:7! -21 £ & H3 43 ¢ -; a 1 0&D $ (*Q(:7X! -21 £ &3
$ (*Q(:7X! -21 £ & H = @ @ B 10.
M]! 1 -; 7 1 -; 0&
V 1 N 1 ! (
3 3 ¢ / £ &-21 0&3 &-21 0& 1 N 1 ! (
1 1 & #$
& ! ( 1
' % &-21 J ( L ""'$9% ')( $ ! D ')( $ ! 1 - &<3 ')( $ ! 1 - 0& H
3 ' % 1 J L "')( $ ! % $ -21
% 3 ¢ a &- -21 0&Y D $ ( Q(87! 1 - £ &<3
$ ( (:7! 1 - £ & H3 43 ¢ a &- -21 0& D $ (*Q(:7X! 1 - £ & $ (* (:7! 1 - £ & H3 3 ¢ -; a 1 & D $ (*Q(:7X! -21 £ &3 $ (*Q(:7X! -21 0& H
;!5
$ (*Q(:7X! -21 £ & H = @ @ B 11.
M]! &-21 &-21 0&3 &-21 &3 £ & &-21 & 43 ¢ C £ & 1 -; 0&3 3 ¢ 1 -; & 3 &-21 0& 43 ¢ 1 -; &0 1 N
1 ! ( 1 &
& &
#$
&-21 0& &-21 0& 43 ¢ 1 -; 0& 3?43 ¢ 1 -; &0 1 N
1 !
3 ¢ 1 -; & 3 ¢ 1 -; 0& 1 N & ! (
1
7 &-21 0&3 43 ¢ 1 -; &0 N 7#7 ! (
1
3 ' % 1 J ( L
"')( $ ! % $ -21
% &- -21 0& 43 ¢ a -; a 1 & Y D $ 5 P5 ! -21 £ &<3 $ 5 P5 ! -21 £ & H 3 ¢ a -; a 1 0&3 &- -21 0& Y D $ 5 P5 ! -21 £ &
$ 5 P5 ! -21 £ & H = @CB 12.
M! &-21 3 &-21 & C £ & 0 &-21 0& 43 ¢ 1 -; 0& 43 ¢ 1 -; 0& ; ;
+ ( L1 3 &-21 0& 43 ¢ 1 -; &<3 &-21 &0 1 N
1 ! ( 1 & & &
#$
3 &-21 0&3 &-21 0& 43 ¢ 1 -; 0& 43 ¢ 1 -; &0 1 N 1 ! (
1 1 & #$
& &-21 & &-21 &3 3 ¢ 1 -; 0& 43 ¢ 1 -; &0 1 N
& ! ( 1
1
3 ' % 1 J L "')( $ ! % $ -21
% Y 43 ¢ &- -21 & 3 ¢ -; a 1 & Y D $ 5 5 ! -21 £ &
$ 5 5 ! -21 £ & H M3 ¢ &- -21 0&3 43 ¢ -; a 1 0& Y D $ 5 P5 ! -21 £ &<3 $ 5 P5 ! -21 £ & H = @CB
13.
M]! 1 -; 43 ¢ a 1 &-21 0&! 3 ¢ a 1 M]! &-21 ¤ 14.
M! 1 -; 43 ¢ a 1 &-21 0& 43 ¢ M! &-21 ¤ 1.18. The Legendre Polynomials Pn(z)
1.18.1. Derivatives with respect to the argument
1. j S 0fI& £ 3 ¢ f a 1 S -; fI& = > 'I@CB 2.
5n'! % J 1 L ' (*'! % F m G ¢ 3hf9 O -; 1 -;SEa fI& = > 'I@CB
;9:
& ' 3. -21 ¢ 3hf & &-21 j f;/ 0
43 ¢ F 7 G f 63hf -;&-21 j F 7m G ¤ 4. DM Q3hf &-21 j F m GIH
43 ¢ F 7G f9 Q3hf9 O -;&-21 j F m G ¤ 5. a 1 ¢ 3hf9 & &-21 j a 1 f / 0
3 ¢ O'65_7 F 7 G f9 a 1 -;&-21 ¢ 3hf9 O& -;&-!&4 j F 7m G ¤ 6. D - + a 1 . Q3hfI &-21 j F 5 m m G9H
F 7 G f -;&-21 63hfI -21 j F " m G ¤ 7. D fg3h& &-21 j F 5 m m G9H
43 ¢ F 7 G f fT3h& -21 j F" m G ¤ 8. DM S fg3h& &- S -21 j S F 5 m m GIH
% ( '2! % f + S -; . 0fg3h& - S -21 j S -; F 5 m m G = > 'I@ B 9. DM - + SEa 1 . 0fg3h& SEa j S F 5 m m GIH
43 ¢ 5n'! % % f - + SEa a 1 . fg3h& S j SEa F 5 m m G ¤ 10. D 0Q3hfI 2j F 5 m m GIH f D j F" m G9H ¤ 11. D 0 3hfI& S j S F E( m ( m GIH
( ! "'C7 ( ! ' ! % (* '! % 0 3hfI& + S -; . j S -; F O( m ( m G = > 'I@CB 12. D &- S -21 0fg3h& S j S F m (8 m ( m GIH " ' m '
J 1 ( L' ! % ( O'! % - S -21 fg3h& + S -; . j S -; F m (8 m ( m G = > 'I@CB 13. ED 0 3hf - + SEa 1 . j S F ( m GIH
43 ¢ 5n'! % % 3hf9 - + SEa a 1 . j SEa F ( m G ¤ ; <
+ ( L1 14. ED 0fg3h& -21 j F#" ¢ 3 m GIH
F 7 G 0fg3h& -;&-21 j )* ( m, ¤ 15. a 1 D a 1 j a 1 F " ¢ 3 m GIH
3 ¢ 'g5 7 F 7 G 63hf9 -;&-21 j )+* E( m-, ¤ 16. k &-21 0fI)3 ¢ -21 j / ¢ 3 fI
F 7 G -;&-21 fI 3 ¢ -I&-21 j F 7 7E( m G ¤ 17. D &-21 ¢ 3hfI& j F 7 7( m GIH
3 ¢ F 7 G -;&-21 j / ¢ 3hfI ¤ 18. ED 0 3hf S j S F ( m GIH
! % ( O'! % 3hf9 S -; j S - F ( m G = > 'I@ B 19. a 1;D 0 3hf9 O S j S:F ( m GIH
! % ( O')(:7X! % 3hf S -;&-21 j S - &-21 F ( m G = > 'g5 7R@CB 20. ED -;&-21 0f9 3h j F m m (8 GIH
43V F 7 G f - &-21 f9 3h j F m m (8 G ¤ 21. D &- S -21 0f 3h S j S F m m (8 GIH
! % (* '! % f9 - S -21 0f9 3h S -; j S - F m m (8 G = > 'I@CB 22. a 1;D &- S 0f9 3h O S j S:F m m (8 GIH 3 ! % (* ')(:7X! % f9 a 1 - S -21 0f9 3h S -;&-21 j S - &-21 F m m (8 G= > '65_7R@CB
23. &-21 ¢ 3 f9 & a 1 < ¢ 3hf9 & &- S -21 j S 0f;/ ! % (* '! % F m G -21 ¢ 3hf9 & - S -21 j S - 0f;/ = > 'I@CB
;9L
) ' * ' 24. ED 0Q3hf9 a 1 ED S 0Q3hf9 &- S -21 j S:F m GIHOH
! % (* '! % F m G S -; 63hf9 - S -21 j S - F m G = > 'I@CB 25. D P D fT3h& S j S F m 5nm (* G9HOH
D % (*'! % H fg3h& S -; j S -; F m 5 m (8 G = > 'I@CB 26. D P D SEa fg3h& - S -21Oj S F m 5 m (8 GIHOH
D 5n'! % % H f S fg3h& - S -;&-21 j SEa F m 5 m (8G ¤ 27. ED PED fT3h& - S -21 j S F m 5 m (8GIHOH
D 5:'! % % H fg3h& - S -;&-21Oj SEa F m 5 m (8 G ¤ 1.19. The Chebyshev Polynomials Tn(z) and Un(z)
1.19.1. Derivatives with respect to the argument
1. ST fI&0 £ &-21 3 ¢ f S -; 0fI& = > '8>?7A@CB 2. EDM -1 ] fg3h& F m 5 m (8GIH 3 ¢ F 7G -;&-21 0fg3h& ¤ 3. 0f9 3h S S ) m 5 m (8 ,
! % ( O'! % f9 3h S -; S -; ) m 5 m (8 , = > 'I@ B 4. &- S -21 0 3hf9 S S ) 5 m ( m , ! % ( O'! % f9 Y - S -21 3 f9 S -; S -; ) 5 m ( m , = > 'I@CB 5. D0 f 3h S S F m m (8 GIH
! % (* '! % 0f 3h S -; S - F m m (8 G = > 'I@ B 6. a 1;D0 f9 3h S S:F m m (8 GIH
! % (*O')(:7X! % ; f9 3h S -;&-21 S - &- F m m (* G = > 'g5 7R@CB ;9P
+ ( P1 7. ED0 f9 3h SEa 1 SEa 1 F m m (8 GIH
5_7! % (* 'g5 7! % f9 3h O S -; a 1 S - a 1 F m m (8 G = > 'I@CB 8. ED0 3hf9 O - S S:F ( m G9H
3 5n')(:7X! % (:7X! % 3hf9 - + SEa . SEa F ( m G = 5n'*>?7R@CB 9. EDM &- S -21 3hf9 S S F ( m GIH
! % (*O'! % f9 - S -21 3 f9 S -; S - F ( m G = > 'I@CB 10. a 1;D &- S 0 3hf9 O S S F ( m GIH 3 ! % (* ' (87! % f9 a - S -21 3hf9 S -;&-21 S - &- F ( m G= > '65_7R@CB
11. ED &- S - 0 3hf9 SEa 1 SEa 1 F ( m GIH 5_7X! % ( O'65_7X! % f9 - S - 0 3hf9 S -; a 1 S - a 1 F ( m G= > 'I@ B
12. ED &-21 PED fT3h& S S:F m 5 m (8GIHOH £ - ! % (*O'! % -21 ] fg3h& S -; S -; F m 5nm (* G = > 'I@ B
13. D a 1 D &- S -21 fg3 S S F m 5 m (8 GIHOH ! % (* '! % F m G - S -21 fg3h& S -; S -; F m 5 m (8G = >_'I@CB
14. ED a 1 PED -21 0fg3h& - S S8F m 5 m (8GIHOH £ - £ fT3h - S -; SEa F m 5 m (8 G ¤
15. ED &-21 PED SEa &-21 fT3h& - S S F m 5 m (8 GIHOH £ - £ f S -21 0fg3h& - S -; SEa F m 5 m (8 G ¤
16. ST0fI& £ fI a 1S -; 0fI& = > 'I@CB : N
Tn(z)
Un(z)
17. ; f9 3h S S ) m 5n m (* , 5n ! % (* '65n ! % ;0f9 3h S -; S -; ) m 5 m (8 , = > 'I@ B
18. &- S - f9 3h S S ) m 5n m (* , 5nO! % (* '65nO! % f - S - 0f 3h S -; S -; ) m 5 m (8 , = > 'I@CB
19. D;0f 3h S S F m m (8 G9H 5_7! % (*O'65 7! % ; f9 3h O S -; S - F m m (* G = > 'I@ B
20. ED;0f9 3h SEa 1 SEa 1 F m m (* G9H 5:O! % (*O'65n ! % ; f9 3h S