mathematics formula 1
TRANSCRIPT
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8/4/2019 Mathematics Formula 1
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Differential Calculus0)( =cd
1= nn nxx
xx cossin =xx sincos =
xx2sectan =
xecx2
coscot =xxx tan.secsec =
xecxecx cot.coscos =xx /1log =
xxee =
xhxh )cos()sin( =
xhxh )sin()cos( =
xhxh )(sec)tan(2
=xhecxh )(cos)cot(
2=
xhxhxh )tan(.)sec()sec( =
hxhecxhec )cot(.)(cos)(cos =
)1(/1)(sin 21 xx =
)1(/1)(cos 21 xx =
)1(/1)(tan 21 xx +=
)1/1)(cot 21 xx +=
))1(/(1)(sec 21 = xxx
))1(/(1)(cos 21 = xxxec
)1(/1)(sinh 21 xx +=
)1(/1)(cosh 21 = xx
)1/(1)(tanh21
xx =
)1/(1)(coth21
xx =
)1(/(1)(sec 21 xxxh =
)1(/(1)(cos 21 += xxxech
Integral Calculus
cnxdxx nn ++= + )1/(1
cxdx += )/1(logcedxe xx +=
cedxe xx +=caexdxe axa +=
)/(
cxxdx += cossincxxdx += sincoscxxdx += tansec
2
cxxdxec += cotcos2
cxxdxx += sectan.seccecxxdxecx += coscot.cos
2
coshsinh
xx eeor
cxxdx
+=
2/)(
sinhcosh
xx eeor
cxxdx
+
+=
cecxxdxec +=
cos2cos
cxxdx +=+ 12
sin)1(/
cxxdx += 12
cos)1(/
cxxdx +=+ 12
tan)1/(
cxxdx +=+ 12 cot)1/(
cxxxdx += 12 sec)1/(
xecxxdx 12 cos)1/( =
cxxxor
cxxdx
+++
+=+ |1|log
sinh)1/(
2
12
cxxxor
cxxdx
++
+= |1|log
cosh)1/(
2
12
||log
log2
1
22
22
axxor
ax
ax
aax
dx
+
+
=
xa
xa
axa
dx
+=
log21
)(22
a
x
axa
dx 122 tan
1
)(
=+
sec
),(cosh 122
ax
a
x
ax
dx
=
=
tan),(sinh
||log
1
22
22
axa
xor
axxxa
dx
=
++=+
ca
x
xa
dx+=
)(sin 122
ca
xaaxx
dxax
+
=
)(cosh22
)
1222
22
cxax
a
xa
dxxa
++
+
=+
2)(sinh
2
)
221
2
22
cxax
a
xa
dxxa
+
+
=
2)(sinh
2
)
221
2
22
Conditions for f(z) to be analytic
1.The 4 partial derivatives yxyx VVUU .,, exist.
2. yxyx VVUU .,, are continuous.
3.CR equation satisfies at every point in the arc.
Harmonic function
Maclaurins series:
If0
z is the origin,then
Isolated singularityIf f(z) has no other singularity in in the
neighbourhood of 0Z ,it is isolated
singularityRemovable singularity
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8/4/2019 Mathematics Formula 1
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Laplace equation in 2 dimensions;Step1
02
2
2
22 =
+
=
yx
Step2: +== )( dxudxudvv xyMilne Thomson Method
)]0,()0,([)( zizuzf yx =Unit V
Cauchy Theorem:Let f(z) be an analytic function defined at all pointsinside and on a simple closed curve,C
then, 0)( =c
dzzf
Cauchy Integral Formula:
Let f(z) be an function which is analytic insideand on a simple closed curve,C,then
=c
zz
dzzf
izf
)(
)(
2
1)(
0
0
Cauchy Residue Theorem:Let f(z) be an function which is analytic insideand on a simple closed curve,C,except for a finitenumber of singular points,then
));((Re2)(1
=
=n
ic
zizfsizf
)___(2 r e s i da l lo fs u mi=
n
n
n
zzn
zfzz
zf
zzzf
zfzf
)(!
...)(!2
)(''
)(!1
)(')()(
0
0
0200
0
0
0
+
++=
=
Taylors series:Let f(z) be an analytic inside a circle,C with center at
0Z .Then f(z) can be represented as a power series
0zz as shown below
n
n
n
zzn
zfzz
zf
zzzf
zfzf
)(!
...)(!2
)(''
)(!1
)(')()(
0
0
02
0
0
0
0
0
+
++=
=
Laurents series:
n
n
n
n
n
n zzazzbzf )()()( 00
0
1
+=
=
=
were
dz
zf
ib
c
inn +=
)(
)(
2
1
0
d
z
f
ia
c
nn +=
2
1
0 )(
)(
2
1
1||......1)1(321
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8/4/2019 Mathematics Formula 1
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TrigonometryNegative Angle
xx sin)sin( = ; xx cos)cos( =
)tan()tan( xx =Pythagorean
1cossin 22 =+ xx ; xx22
sectan1 =+ ;
xecx 22 coscot1 =+ .Sum & Difference of Angles
BABABA sincoscossin)sin( +=+BABABA sincoscossin)sin( =BABABA sinsincoscos)cos( =+BABABA sinsincoscos)cos( +=
BA
BABA
tantan1
tantan)tan(
+=+
BA
BABA
tantan1
tantan)tan(
+=
Double Angle
)2/(tan1
)2/tan(2sin
2A
AA
+=
)2/(tan1
)2/(tan12cos
2
2
A
AA
+
=
AAA cossin22sin =
AA2
sin212cos = ; AA 22 sincos ;
1cos2
2
A
A
AA
2tan1
tan22tan
=
AAA3
sin4sin33sin =
AAA cos3cos43cos 3 =
A
AAA
2
3
tan31
tantan33tan
=
Product to sum
)]sin()[sin(2/1cossin BABABA ++=)]cos()[cos(2/1coscos BABABA ++=)]cos()[cos(2/1sinsin BABABA +=
BA
BABA
cotcot
tantantantan
+
+=
Sum and Difference of functions
)]
(2
1cos).(
2
1[sin2sinsin BABABA ++=+
)](2
1cos).(
2
1[sin2sinsin BABABA +=
(21cos).(
21[cos2coscos BABABA +=+
(2
1sin).(
2
1[sin2coscos ABABA +=
BA
BABA
coscos
)sin(tantan
+=+
BA
BABA
coscos
)sin(tantan
=
Logarithms
1log =cc ; PcP
c=log ;
baba ccc loglog).(log +=baba ccc loglog)/(log =
baba cc == loglog
xmx am
a loglog =
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8/4/2019 Mathematics Formula 1
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Eulers Theorem on Homogeneousfunction
If u=u(x,y) is a homogeneous function of
degree n,then, nux
u
yx
u
x =
+
..Total differential CoefficientIf u=f(x,y),where x=g(t) & y=f(t),then
t
y
y
f
t
x
x
f
dt
du
+
= ..
Implicit FunctionIf an implicit function f(x,y)=0;then
yf
xf
dx
dy
=
/
/
Taylors series ExpansionThe expansion of f(x,y) about the point (a,b) is
);(
2
2
22
2
22
;(
)],(.2
,(.),(.[!2
1
)],(.),(.[
),(),(
byax
ax
yxfyx
hk
xfy
kyxfx
h
yxfy
kyxfx
h
bafyxf
==
=
+
+
+=