2 2 2 2 2 2 important concept and formulae algebraic ...trigonometric ratios: ' abc is a right...
TRANSCRIPT
Important Concept and Formulae
Algebraic Identities:
,
, ,
2 22 2 2 2 2 2,
22 2 2 2
3 33 2 2 3 3 3 3 2 2 3 3 3
3 3 2 2 3
1) 2 2) 2 3) ,
4)( )( ) , 5) 2 2 2 ,
6) 3 3 3 7) 3 3 3
8) , 9)
a b a ab b a b a ab b a b a b a b
x a x b x a b x ab a b c a b c ab bc ca
a b a a b ab b a ab a b b a b a a b ab b a ab a b b
a b a b a ab b a
3 2 2
3 3 3 2 2 2 4 4 2 2
,
10) 3 ( 2 2 2 ), 11) ( ).
b a b a ab b
a b c abc a b c a b c ab bc ca a b a b a b a b
Laws of Logarithms: 1) If log , 0, 1, 0, then .
2) log ( ) log log , , , 0, 1, 3) log ( / ) log log , , , 0, 1,
log4) log log , , , 0, 1, 5) log , , , 0, 1,
log
16) log , , 0, 1, 1, 7) log 1
log
y
a a a a a a
n aa a n
a
n am
y x a a x x aa
mn m n m n a a m n m n m n a a
mm n m m n a a m m n a a
n
m m n m nn
log log, ,
0, 0, 1, 8) log 1, 0, 1,
9) , 0, 1, 10) , 0, 2.718.
a
x xa ex x
a a a a a
a a x a e e x e
Laws of Indices:
, , ,
,
)
0
1) , ( 3) 4)
15) , 6) , 7) 8) 1.
2)m
m n m n m n mn m m m m nn
mm mmm
mm
aa a a a a a b a b a
a
aa a ba a
ab bb a
Trigonometry 0
00
, ,
1)The unit of angle measureisdegreeand it is denoted by .
2)The unit of angle measureis radian and it isdenoted by .
180 1803) π radian =180degree i.e. 180 1radian = degree i.e. 1
180radian = degree i.
C C
c
,
0
0 0
0 0 0
0
180e.
4)1degree = radian i.e.1 degree = radian i.e.180 180 180 180
5) 1 57 17 48 57.3 (approx.), 1 0.01745 (approx.)
6)1degree 60minutes i.e.1 60 , 1minute = 60
C
C C
C c
x x x x
seconds i.e.1 =60 .
Trigonometric Ratios: ABC is a right angled triangle as shown in the figure. AB is adjacent side, BC is opposite side and AC is
hypotenuse, then
Oppositeside Adjacent side Oppositeside1)sinθ= , 2)cosθ= , 3) tanθ= ,
Hypotenuse Hypotenuse Adjacent side
Hypotenuse Hypotenuse Adjacent side4)cosecθ= , 5)secθ= , 6)cot θ =
Oppositeside Adjacent side Op
BC AB BC
AC AC AB
AC AC
BC AB
.positeside
AB
BC
Inter-relation between trigonometric ratios: 1 1 1 sinθ cosθ
1)cosecθ= , 2)secθ= , 3) tanθ= , 4) tanθ= , 5)cotθ=sinθ cosθ cot θ cosθ sinθ
2 2 2 2 2 26) sin cos 1, sin 1 cos , cos 1 sin
2 2 2 2 2 27) 1 tan sec , sec tan 1, tan sec 1 2 2 2 2 2 28) 1 cot sec , sec cot 1, cot sec 1co co co
Trigonometric ratios of negative angles:
1) sin sin , cos cos , tan tan ,
2)cosec cosec , sec sec , cot cot .
Trigonometric functions of standard angles:
Angle
Ratio
00 030
( / 6)c
045
( / 4)c
060
( / 3)c
090
( / 2)c
0180
0270
(3 / 2)c
0360
(2 )c
sin 0 1
2
1
2 3
2
1 0 -1 0
cos 1 3
2
1
2 1
2
0 -1 0 1
tan 0 1
3
1 3 0 0
cosec 2 2 2
3
1 -1
sec 1 2
3 2 2 -1 1
cot 3 1 1
3
0 0
Signs of trigonometric functions: Quadrants
Trigonometric
ratios
I I I III IV
sin + + - - cos + - - + tan + - + -
Trigonometric functions of addition and subtraction: 1) sin( ) sin cos sin cos , 2) sin( ) sin cos sin cos ,
3) cos( ) cos cos sin sin , 4) cos( ) cos cos sin sin ,
tan tan tan tan5)tan( ) , 6)tan( ) ,1 tan .tan 1 tan .tan
7)sin( )sin( ) s
A B A B B A A B A B B A
A B A B A B A B A B A B
A B A BA B A BA B A B
A B A B
2 2 2 2in sin , 8)cos( )cos( ) cos cos .A B A B A B A B
Trigonometric functions of allied angles
Angle
Ratio
2
2
3
2
3
2
2 2
sin sin cosθ cosθ sin sin cosθ cosθ sin sin
cos cosθ sin sin cosθ cosθ sin sin cosθ cosθ
tan tan cotθ cotθ tan tan cotθ cotθ tan tan
cosec cosecθ secθ secθ cosecθ cosecθ secθ secθ cosecθ cosecθ
sec secθ cosecθ cosecθ secθ secθ cosecθ cosecθ secθ secθ
cot cotθ tan tan cotθ cotθ tan tan cotθ tan
Trigonometric ratios of large angles: 1) When n is an even integer then,
) sin sin , ) cos cos , ) tan tan , etc.2 2 2
2) When n is an odd integer then,
) sin cos , ) cos sin , ) tan cot , etc.2 2 2
a n b n c n
a n b n c n
The correct sign is to be chosen depending the quadrant in which the angle lies, using the sign convention.
Trigonometric functions of double and triple angles: 2 2 2 2
2
2 2 2
2 2 2 2
1) sin 2 2sin cos , 2)cos 2 cos sin 1 2sin 2cos 1,
2 tan 1 tan 2 tan3) sin 2 , 4) cos 2 , 5) tan 21 tan 1 tan 1 tan
1 cos 2 1 cos 26)1 cos 2 2sin , sin , 7)1 cos 2 2cos , cos ,2 2
8) 1 sin 2 (cos
2 2, ,
33 3
3
sin ) 9) 1 sin 2 (cos sin )
3tan tan10) sin 3 3sin 4sin , 11) cos3 4cos 3cos , 12) tan 3 .1 3tan
Trigonometric functions of half angles: 2 2 2 2
2
2 2 2
2 2
1) sin 2sin( / 2)cos( / 2), 2)cos cos ( / 2) sin ( / 2) 1 2sin ( / 2) 2cos ( / 2) 1,
2 tan( / 2) 1 tan ( / 2) 2 tan( / 2)3) sin , 4)cos , 5) tan
1 tan ( / 2) 1 tan ( / 2) 1 tan ( / 2)
6)1 cos 2sin ( / 2), 7)1 cos 2cos ( /
.2 2
,
2),
8)1 sin cos( / 2) sin( / 2) 9)1 sin cos( / 2) sin( / 2)
Factorisation Formulae:
1) sin sin 2sin cos , 2) sin sin 2cos sin ,2 2 2 2
3) cos cos 2cos cos ,2 2
C D C D C D C DC D C D
C D C DC D
4) cos cos 2sin sin 2sin sin .2 2 2 2
C D D C C D C DC D
Defactorisation Formulae: 1) 2sinAcos sin( ) sin( ), 2) 2cosAsin sin( ) sin( ),
3) 2cosAcosB cos( ) cos( ), 4) 2sinAsin cos( ) cos( ).
B A B A B B A B A B
A B A B B A B A B
Inverse trigonometric functions: 1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1
1) sin ( ) cosec (1/ ), cosec ( ) sin (1/ ),
cos ( ) sec (1/ ), sec ( ) cos (1/ ),
tan ( ) cot (1/ ), cot ( ) tan (1/ ).
2) sin ( ) sin ( ), cos ( ) cos ( ),
tan ( ) tan ( ), cot ( )
x x x x
x x x x
x x x x
x x x x
x x x
1
1 1 1 1
1 1 1
1 1 1
1 1 1 1 1 1
cot ( ),
cosec ( ) cosec ( ), sec ( ) sec ( ).
3)sin (sin ) , cos (cos ) , tan (tan ) ,
cot (cot ) , cosec (cosec ) , sec (sec ) .
4)sin cos / 2, tan cot / 2, cosec sec
x
x x x x
x x x x x x
x x x x x x
x x x x x x
1 1 1
/ 2.
5) If 0, 0and 1then, tan tan tan .1
x yx y xy x y
xy
1 1 1
1 1 1
6) If 0, 0and 1then, tan tan tan .1
7) If 0, 0 then, tan tan tan .1
x yx y xy x y
xy
x yx y x y
xy
Trigonometric functions of angles of triangle: 1) In any ABC,
( )sin( ) sin , ( )sin( ) sin , ( )sin( ) sin ,
( )cos( ) cos , ( )cos( ) cos , ( )cos( ) cos .
i A B C ii B C A iii C A B
iv A B C v B C A vi C A B
2) In any ABC,
( )sin cos , ( )sin cos , ( )sin cos ,2 2 2 2 2 2
( )cos sin , ( ) cos sin , ( ) cos sin .2 2 2 2 2 2
A B C B C A C A Bi ii iii
A B C B C A C A Biv v vi
Properties of a triangle: If A,B,C are angles ofΔABC and a,b,c are thelengths of the sides oppositetothe angles A,B,C respectivelythen,
1)Sine rule:sin sin sin
a b cA B C
2 2 22 2 2
2 2 22 2 2
2 2 22 2 2
2)Cosinerule: cos 2 .cos2
cos 2 .cos2
cos 2 .cos2
3)Projection rule: cos cos , cos cos , cos cos
b c aA or a b c bc Abc
c a bB or b c a ca Bca
a b cC or c a b ab Cab
a b C c B b c A a C c a B b A
4)Tangent rule(Napier's rule):
tan cot , tan cot , tan cot .2 2 2 2 2 2
A B a b C B C b c A C A c a Ba b b c c a
Quadratic Equations 2
2
2 2
1)Theequation of the form 0, where a,b,c are real numbers and 0 is called quadratic equation.
42)The solution of the quadratic equation is .
2
3) 1 is called an imaginary number. 1
ax bx c a
b b acx
a
i j i j
Progression and Series
1
2 2 2 2 2
1
3 3 3 3 3
( 1)1)Sum of the first n natural numbers: 1 2 3
2
( 1)(2 1)2)Sum of the squaresof thefirst n natural numbers: 1 2 3
6
3)Sum of the cubesof thefirst n natural numbers: 1 2 3
n
rn
r
r
n nn r
n n nn r
n n
( (22 2
1
1) 1)
4 2
n n n n n
Binomial Theorem
,0 1
Factorial Notations:
1) n factorial is denoted by n!, 2) ! 1 2 3 ( 2) ( 1) , 3)0! 1, 4)1! 1
Combinations:
!1) where ( ) , 2) 3) 1, 4) .
!( )!n n n n n n
r r n r n
n n n n
nC r n C C C C C n
r n r
1 2 2 3 31 2 3
1 2 2 3 31 2 3
BinomialTheorem:
1) If a,b are real numbersand n is non-negativeinteger then,
i) ( )
ii) ( ) ( 1)
n n n n n n n n n n r r nr
n n n n n n n n r n n r rr
a b a C a b C a b C a b C a b b
a b a C a b C a b C a b C a b
.
2 31 2 3
2 31 2 3
( 1)
2) (1 ) 1
3) (1 ) 1 ( 1)
n n
n n n n n
n n n n n n
b
x C x C x C x x
x C x C x C x x
Limits ,
1
0 0 0
1/
0
0 0
sin tan1) lim 2) lim 1, 3) lim (cos ) 1, 4) lim 1,
5) lim (1 ) , where 2.718, 6) lim (1 1/ ) , where 2.718,
log(1 ) log(1 )17) lim 0, ( 0) 8) lim 1, 9) lim , wher
n nn
x a
x x
xx
nx x x
x an a
x a
x e e x e e
x mxn m
x xx
0 0
e mis constant,
1 110) log , ( 0, 1) 11) log 1.
x x
ex x
a ea a a elim limx x
Complex Number
Definition - A number of the form z = x + iy where x and y are real numbers and i 1 is called a
complex number.
Note : 1) x is called the real part of z and it is denoted by R(z) or Re(z) and y is called the imaginary part of
z and it is denoted by I(z) or Im(z).
2) i 1 2i 1 3) i ,j and k are imaginary numbers and i j k 1 .
Equality of two complex numbers – Two complex numbers are said to be equal if and only if their real
parts are equal and imaginary parts are equal .
Conjugate of complex number- Two complex numbers which differ only in the sign of imaginary part are
said to be conjugate of each other.
e.g. a + ib and a – ib are conjugates of each other.
Algebra of complex numbers – Let 1z a ib and 2z c i d be two complex numbers where
, , ,a b c d R and 1i . Then addition, subtraction, multiplication and division are performed as
follows:
1) Addition: 1 2 ( ) ( )z z a ib c id ( ) (b )a c i d
2) Subtraction: 1 2 ( ) ( )z z a ib c id ( ) (b )a c i d
3) Multiplication: 1 2 ( )( )z z a ib c id ( ) ( )a c id ib c id
2ac iad ibc i bd 2 1]( ) ( ) [ac bd i ad bc i
4) Division: 1
2
z a ib
z c id
a ib c id
c id c id
2 2 2
2ac iad ibc i bd
c i d
2 2
( ) ( )ac bd i bc ad
c d
2 1][ i
Modulus and argument(amplitude) of a complex number:
1) If z = x + iy is a complex number, then the modulus of z is defined to be2 2x y and is denoted by r
or z or mod z. 2 2r z x y
2) If z = x + iy is a complex number, then the argument of z is defined as 1tan ( / )y x and is denoted by
or arg(z) or amp(z) . 1tan ( / )y x
Powers of a complex number i :
2 2 2) )2 3 2 4Wehave i 1, 1 1 ( ( 1 1i i i i i i i i
Polar form of a complex number:
If z = x + iy is a complex number, then the polar form of complex number z is given by(cos sin )z r i
Working rule to find amplitude :
If z = x + iy is a complex number, then1tan
y
x and we have following cases:
1) If 0, 0(Iquadrant), then . 2) If 0, 0(IIquadrant), then - .
3) If 0, 0(IIIquadrant), then . 4) If 0, 0(IVquadrant), then 2 .
x y x y
x y x y
Exponential form of a complex number:
If z = x + iy (Cartesian form) is a complex number, then z=rei is called the exponential form of a
complex number.
Note: cos sin , 2) cos sin1) e ei ii i
Circular Functions: 1)cos , 2)sin2 2
e e e ei i i i
i
These formulae are called as Euler’s Exponential Functions.
Hyperbolic Functions: 1)cosh , 2)sinh2 2
e e e e
Relation between circular and hyperbolic functions:
1)sini sinh
2)cosi cosh
3) tani tanh
i
i
1)sinhi sin
2)coshi cos
3) tanhi tan
i
i
De Moivre’s Theorem:
If n is any real number, then one of the values of (cos sin ) iscos sinni n i n .
(cos sin ) cos sinni n i n
Roots of a complex number:
If z = x + iy is a complex number, then the polar form of a complex number is,
(cos sin )z x iy r i
Taking n th root on both sides,
1/ 1/ 1/ 1/ 1/ 1/
1/
( ) (cos sin ) [cos(2 k ) sin(2 k )]
[cos(2 k ) / n sin(2 k ) / n] [ By De Moivre'sTheorem]
where k 0,1,2 n 1
n n n n n n
n
z x iy r i r i
r i
We put above values to find the roots.
Derivatives Derivative of standard functions:
,1
2
2
2
1/( ) ( ) 1 11) 2) , 3) 1, 5) 0
2
log( ) ( ) 16) log , ( 0) 7) , 8) ,
sin cos (tan )9) cos , 10) sin , 11) sec ,
cot12) co
, 4) where k is constant.
sec , 1
nn
x xx x
d x d x d kd x d xn x
dx dx dx dx dxx x
d xd a d ea a a e
dx dx dx x
d x d x d xx x x
dx dx dx
d xx
dx
1 1 1
22 2
1 1 1
2 2 2
sec cosecx3) sec tan , 14) cosecx cot ,
(sin ) (cos ) (tan )1 1 115) , 16) , 17) ,
11 1
(cot ) (sec ) (cosec )1 1 118) , 19) , 20) .
1 1 1
d x dx x x
dx dx
d x d x d x
dx dx dx xx x
d x d x d x
dx dx dxx x x x x
Integration Integration of standard functions:
1
2
2
1) where 1, 2) ,1
3) log where 0, 4) where 0 and 1, 5) ,log
6) sin cos , 7) cos sin , 8) sec tan ,
9) cosec cot , 10) secx tan secx
nn
xx x x
xx dx c n dx x c
n
dx ax c x a dx c a a e dx e c
x a
xdx x c xdx x c xdx x c
xdx x c xdx c
1 1
2
, 11) cosecx cot cosec x ,
12) tan log sec , 13) cot log sin ,
14) sec log sec tan log tan , 15) cosec log cosec cot log tan ,2 4 2
16) sin cos , 17)1
xdx c
xdx x c xdx x c
x xxdx x x c c xdx x x c c
dx dxx c x c
x
1 12
1 1
2
1 1 1 12 22 2
1 1
2 2
2 2
tan cot ,1
18) sec cosec ,1
1 119) sin / cos / , 20) tan / cot / ,
1 121) sec ( / ) cosec ( / ) ,
22)
x c x cx
dxx c x c
x x
dx dxx a c x a c x a c x a c
a ax aa x
dxx a c x a c
a ax x a
dx
x a
2 2 2 2
2 2
2 2 2 2
2 22 2 2 2 1 2 2 2 2
log , 23) log ,
1 124) log , 25) log ,
2 2
26)
27) sin ( / ) , 28) log2 2 2 2
dxx x a c x x a c
x a
dx x a dx a xc c
a x a a a xx a a x
duuvdx u vdx vdx dx
dx
x a x aa x dx a x x a c x a dx x a
2 2
22 2 2 2 2 2
2 2 2 2
( ) ,
29) log( ) ,2 2
30) sin [ sin cos ] , 31) cos [ cos sin ]
32) ( ) ( ) ( )
ax axax ax
x x a c
x ax a dx x a x x a c
e ee bx dx a bx b bx c e bx dx a bx b bx c
a b a b
x xe f x f x dx e f x c
Standard Substitutions:
Integrand Substitution
2 2a x sin or cosx a x a
2 2a x tan or cotx a x a
2 2x a sec or cosecx a x a
ora x a x
a x a x
cos 2x a or sin 2x a
a x
x
2 2sin or cosx a x a
a x
x
2 2tan or cotx a x a
( )( ) orx a
x a b xb x
2 2sin cosx a b
Definite Integral Definition of Definite Integral - If f(x) is a function defined on the closed interval [a,b] and
( ) ( )f x dx g x , then ( )b
a
f x dx g(b) – g(a) where ‘a’ is called the lower limit and ‘b’ is called upper
limit of integration.
Note - If f(x) and g(x) are two functions and k is a constant, then
1) ( ) g(x) ( ) g(x)b b b
a a a
f x dx f x dx dx 2) ( ) ( )b b
a a
k f x dx k f x dx
Properties of Definite Integral
1) If the limits of a definite integral interchanged, then the sign of the integral changes. ,
( ) ( )
b a
a b
f x dx f x dx
2) Change of independent variable in the given function does not change the value of definite integral.
( ) ( )
b a
a b
f x dx f t dt
3) If a < c < b, then a definite integral on an interval [a,b] can be expressed as sum of two definite integrals.
( ) ( ) ( )
b c b
a a c
f x dx f x dx f x dx
4)
0 0
( ) ( )
a a
f x dx f a x dx . 5) ( ) ( ) .
b b
a a
f x dx f a b x dx 6)
2
0 0 0
( ) ( ) (2 ) .
a a a
f x dx f x dx f a x dx
0
7) ( ) 2 ( ) if ( ) is even function.
0 if ( ) is an odd function.
a a
a
f x dx f x dx f x
f x
Applications of Definite Integral
1) The area A bounded by the curve y = f(x), X – axis and lying between the lines x = a and x = b is given
by,
( ) .
b b
a a
A ydx f x dx
2) The area A bounded by the curve x = g(y), Y – axis and lying between the lines y = c and y = d is given
by,
( ) .
d d
c c
A xdy g y dy
3) If the region bounded by the curve y = f(x), the x – axis and the lines x = a and x = b is revolved about
the x-axis, then the volume of solid of revolution is given by,
2 2[ ( )] .
b b
a a
V y dx f x dx
4) If the region bounded by the curve x = g(y), the y – axis and the lines y = c and y = d is revolved about
the y-axis, then the volume of solid of revolution is given by,
2 2[ ( )] .
b b
a a
V x dx g y dy
Laplace Transform
2 2 2 2
12 2 2 2
2 2
1 11) (1) where 0, 2)L(e ) where , 3) (sinat) , 4) (cosat) ,
!5) (sinhat) where , 6) (coshat) where , 7) ( ) ,
8) ( sin t) where , 9) ( cos t)( ) ( )
at
nn
at at
a sL s s a L Ls s a s a s a
a s nL s a L s a L tss a s a
b s aL e b s a L e bs a b s a
2 2
2 2 2 2
1
1 1
1 1
2 2 2 2
where ,
10) ( sinh t) where , 11) ( cosh t) where ,( ) ( )
!12) ( ) where ,( )
1 113) ( ) 1 where 0, 14) e where ,
1 115) sinat, 16)
at at
at n
n
at
s ab
b s aL e b s a L e b s as a b s a b
nL e t s as a
L s L s as s a
sL Las a s a
1 1
2 2 2 2
1
1
cosat,
1 117) sinhat, 18) coshat,
119)!
n
n
sL Las a s a
tLns
Fourier Series
Definition of Fourier Series: The Fourier Series for the function f(x) in the interval a < x < a + 2l is given
by,
0
1 1
( ) cos sin2
n n
an x n x
f x a bn nl l
where 2
0
1( )
a l
aa f x dx
l
21( )cos( / )
a l
n aa f x n x l dx
l
, 1,2, ,n
21
( )sin( / )a l
na
b f x n x l dxl
, 1,2, ,n
The formulae for 0
a , na and nb are called Euler’s formulae.
The real constants 0 1 2 1 2, , , , , ,n na a a a b b b are called Fourier constants of the
function f(x).
Note: 1) The Fourier expansion of the function f(x) in the interval (0,2l) is given by,
0
1 1( ) cos sin
2 n nn n
a n x n xf x a b
l l
where 2
0 0
1( )
la f x dx
l
2
0
1( )cos( / )
l
na f x n x l dxl
, 1,2, ,n
2
0
1( )sin( / )
lb f x n x l dxn l
, 1,2, ,n
2) The Fourier expansion of the function f(x) in the interval (0,2 ) is given by,
0
1 1( ) cos( ) sin( )
2 n nn n
af x a nx b nx
where 2
0 0( )
1a f x dx
2
0
1( )cos( )na f x nx dx
, 1,2, ,n
2
0
1( )s ( )nb f x in nx dx
1,2, ,n
3) The Fourier expansion of the function f(x) in the interval (-l,l) is given by,
0
1 1( ) cos sin
2 n nn n
a n x n xf x a b
l l
where 0
1( )
l
la f x dx
l
1( )cos( / )
l
nl
a f x n x l dxl
, 1,2, ,n
1( )sin( / )
l
lb f x n x l dxn l
, 1,2, ,n
4) The Fourier expansion of the function f(x) in the interval ( , ) is given by,
0
1 1( ) cos( ) sin( )
2 n nn n
af x a nx b nx
where 0
1( )a f x dx
1
( )cos( )na f x nx dx
, 1,2, ,n
1( )s ( )nb f x in nx dx
, 1,2, ,n
5) sin( ) 0n for all values of n , cos( ) ( 1)nn when n is odd number and cos( ) 1n when n is even
number.
6) If u and v are functions of x, then the Leibnitz rule for successive integration by parts (Generalized rule
of integration by parts) is given by,
1 2 3 4uvdx uv u v u v u v
where dashes denote differentiation and suffixes denote integration.
7) ∫ 𝑒𝑎𝑥𝑠𝑖𝑛𝑏𝑥𝑑𝑥 =𝑒𝑎𝑥
𝑎2+𝑏2 [𝑎𝑠𝑖𝑛𝑏𝑥 − 𝑏𝑐𝑜𝑠𝑏𝑥] + 𝑐
8) ∫ 𝑒𝑎𝑥𝑐𝑜𝑠𝑏𝑥𝑑𝑥 =𝑒𝑎𝑥
𝑎2+𝑏2 [𝑎𝑐𝑜𝑠𝑏𝑥 + 𝑏𝑠𝑖𝑛𝑏𝑥] + 𝑐
Fourier Series Expansion of Even and Odd Function:
1) Even Function: If f(-x) = f(x), then the function f(x) is called even function.
2) Odd Function: If f(-x) = - f(x), then the function f(x) is called odd function.
Note: 1) The Fourier Series of the function f(x) when it is odd function on the interval (-l,l) is given
by,
1
( ) sinnn
n xf x b
l
where 0
0na a and 0
2( )sin( / )
lb f x n x l dxn l
.
2) The Fourier Series of the function f(x) when it is an even function on the interval (-l,l) is given by,
0
1( ) cos
2 nn
a n xf x a
l
where 0 0
2( )
la f x dx
l ,
0
2( )cos( / )
l
na f x n x l dxl
and 0nb .
3) The Fourier Series of the function f(x) when it is odd function on the interval ( , ) is given by,
1
( ) sin( )nn
f x b nx
Where 0
0na a and 0
2( )s ( )nb f x in nx dx
.
4) The Fourier Series of the function f(x) when it is even function on the interval ( , ) is given by,
0
1( ) cos( )
2 nn
af x a nx
where 0 0
( )2
a f x dx
,
0
2( )cos( )na f x nx dx
and 0nb .
Half Range Fourier Series:
1) Half range Fourier Cosine Series of the function f(x) over (0,l) is given by,
0
1( ) cos
2 nn
a n xf x a
l
where
0 0
2( )
la f x dx
l ,
0
2( )cos( / )
l
na f x n x l dxl
2) Half range Fourier Cosine Series of the function f(x) over (0, ) is given by,
0
1( ) cos( )
2 nn
af x a nx
where
0 0( )
2a f x dx
,
0
2( )cos( )na f x nx dx
3) Half range Fourier Sine Series of the function f(x) over (0,l) is given by,
1
( ) sinnn
n xf x b
l
where
0
2( )sin( / )
lb f x n x l dxn l
.
4) Half range Fourier Sine Series of the function f(x) over (0, ) is given by,
1
( ) sin( )nn
f x b nx
where
0
2( )s ( )nb f x in nx dx
.
Fourier Transform Definition of Fourier Transform: The Fourier Transform of the function f(x) is given by,
1( ) ( )
2
isxF s f x e dx
Note: 1) The Fourier Sine Transform of f(x) is given by,
0
2( ) ( )sinsF s f x sx dx
2) The Fourier Cosine Transform of f(x) is given by,
0
2( ) ( )coscF s f x sx dx