Mathematical Physics II-Function of Complex

8/17/2019 Mathematical Physics II-Function of Complex

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Mathematical Physics II

R. Arif WibowoPhysics Department,

Faculty Of Sciences an !echnolo"y Airlan""a #ni$ersity%&&'


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Functions of A (omple) *ariable


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(omple) +umber Re$iew-

• A comple) number has two parts, i.e. a

real an an ima"inary part.

• et a comple) number z, which be written

as / 0 x 1 iy 0 r eiθ . !he real part is x an

the ima"inary one is y .

• !he plottin" of the number is showe by

fi"ure 2 below.


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z 0 x 1 iy

x

y

r

3

4

θ

Fi". 2. Plottin" comple) number in the comple) plane


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Drill

• Fin the real an ima"inary part, con5u"ate, ans6uare absolute of the followin" numbers

2352

ii+−

523

37

i

i

−

−

ii2653

++

52

332

−

−

i

i

2253

ii

−−

i

i

−

+

2

37

23π ie

a- b- c-

- e- f-

"-


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Function of z

• (onsier a simple function of z

f /- 0 z % 0 x+iy -% 0 x % 7 y % 1 % xyi

= u x,y - 1 i v x,y -

• Where : u x,y - 0 x % 7 y %

• An v x,y - 0 % xy

f /- 0 f x+iy - = u x,y - 1 iv x,y -


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Problems

• Fin the real an ima"inary parts of the

followin" functions

2

2

+−

iz

i z

2. /8

%. e/

8. .

9. ln /

5. √z6. . z


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!a:e ;ome Problems

• Fin the real an ima"inary parts of the

followin" functions

i z z +2

3 z z ln

z cosh

∗ z z 2

z

( ) ( ) 3121 2 +−++ z i z i

3ln z iz e

• (areful< cos z an sin z are not u an v

2-

%-

8-

9-

=-

>-

?-

@-

'-


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Analytic Function

• Definition< A function f(x) is analytic or re"ular orholomorphic ormonogenic) in a re"ion of

the comple) plane if ithas a uni6ue- eri$ati$eat e$ery point of there"ion. !he statementf z - is analytic at a point

z 0 aB means that f z - hasa eri$ati$e at e$ery pointinsie some small circleabout z 0 a.

( ) z

f

dz

df z f

z ∆∆

==′→∆

lim0

( ) ( ) z f z z f f −∆+=∆

yi x z ∆+∆=∆

en"an


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!heorem I

• If f z - 0 u x,y - 1 iv x,y -

is analytic in a re"ion,

then in that re"ion y

v

x

u

∂∂

=∂∂

y

u

x

v

∂∂

−=∂∂

!hese e6uation are calle the(auchy7Riemann conition


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Problems

• #se the (auchy7Riemann conition to fin out

whether these functions are analytic

22

+−

iz i z

2. /8

%. e/

8. .

4. √z=. ln /

>. . z


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!a:e ;ome Problems

• #se the (auchy7Riemann conition to fin out

whether these functions are analytic

i z z +2

z z ln

z cosh

∗ z z 2

z

( ) ( ) 3121 2 +−++ z i z i

z lniz e• (areful< cos z an sin z are not u an v

2-

%-

8-

9-

=-

>-

?-

@-

'-


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!heorem II

•

If u x,y - an v x,y - an theirpartial eri$ati$es with respectto x an y are continuous ansatisfy the (auchy7Riemannconitions in a re"ion, then f z -is analytic at all point insie

the re"ion not necessarily onthe bounary-.

x

vi

x

u

z

f

∂∂+

∂∂=

∂∂

!hus fC/ has the same $alue when calculate for approach alon" any strai"ht line.

!he theorem states that it also has the same $alue for approach alon" any cur$e.

Some efinitions< A re"ular point of f/- is a point at which f/- is analytic.

A sin"ular point or sin"ularity of f/- is a point at which f/- is not analytic, It is

calle an isolate sin"ular point if f/- is analytic e$erywhere else insie some

small circle about the sin"ular point .


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Drill

• #sin" the efinition of e/ by its power

series show that C/-e/- 0 e/

• #sin" the efinition of sin / an cos / fin

their eri$ati$es.

• Fin C/-cot /-, if / ≠ nπ


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!heorem III If f/- is analytic in a re"ion R in fi"ure-, then it has

eri$ati$es of all orers at points insie the re"ion ancan be e)pene in a !aylor series about any point /o insie the re"ion. !he power series con$er"es insie thecircle about z o that e)tens to the nearest sin"ular point( in fi"ure-.

Sin"ular pointz o

R

C A function φ),y-which satisfies

aplaces e6uation

is calle aharmonic function.

( ) 0,2

2

2

22 =

∂∂

+∂∂

=∇ y x

y x φ φ

φ !he aplaces e6uation is<


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!his theorem also e)plains a fact about power

series

( ) 64211

1 z z z

z

z f −+−=+

=

When / 0 E i, then f/- an its eri$ati$es becomeinfinite that is, f/- is not analytic in any re"ion

containin" / 0 E i.


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!a:e home problems

Show that the following functions are harmonic

1. x 1 y

%. 8 x %y G y 8

8. (osh y cos )

. e x

cos y !. e "y sin x

>. ln x % 1 y %-

( ) 221 y x

y

+−

22 y x

x

+7

8


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#se theorem III to fin the circle of con$er"ence of

each series of the followin" functions

• ln 2 G /-

• cos /

•

tanh /• 2 G /- G2

• ei/


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!heorem I*

Part I.

If f/- 0 u 1 i$ is analytic in a re"ion, then u an $ satisfy

aplaces e6uation in the re"ion that is, u an $ are

harmonic functions-.

Part II.

Any function u or $- satisfyin" aplaces e6uation in a simply7

connecte re"ion, is the real or ima"inary part of an

analytic function f/-.


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Drill• 2-HPro$e !heorem I*, Part I.

• %-HFin the (auchy7Riemann e6uation in

a polar coorinates

• #sin" results in problem %- an the

metho of problem 2-, show that u an $

satisfy aplaces e6uation in polar

coorinates if f/-0u1i$ is analytic.

• #sin" polar coorinates fin out whetherthe followin" function satisfy the (auchy7

Riemann e6uations. a- ln /, b- /n c- I/I


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(ontour Inte"rals

et ( be a simple close cur$e with a continuously turnin" tan"ents e)cept

possibly at a finite number of points that is, we allow a finite number of

corners, but otherwise the cur$e must be smoothB-. If f(z) is analytic on aninsie C , then

!heorem *(auchys !heorem

( ) 0Caround

=

∫ dz z f

(#hi$ i$ a line in%egral a$ in vec%or analy$i$& i% i$ calle' a con%our in%egral in %he

%heory of complex variale$)


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!heorem *I

(auchys Inte"ral Formula

If f/- is analytic on insie a simple close cur$e (, the $alue of f/- at a point / 0 a

insie ( is "i$en by the followin" contour inte"ral alon" (<

( ) ( )∫ −= dz a z

z f

ia f

π 2

1

(#hi$ i$ Cauchy$ *n%egral ormula. o%e carefully %ha% %he poin% a i$ in$i'e C& if a

wa$ ou%$i'e C, %hen φ (z) woul' e analy%ic everywhere in$i'e C an' %he

in%egral woul' e zero y Cauchy$ #heorem.)


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(auchys Inte"ral Formula

In eneral

Jy ifferentiatin" we ha$e,

( ) ( )

( )∫ −=′ dw z ww f

i z f

22

1

π

Jy ifferentiatin" n times, we obtain

( ) ( )

( )∫ +

−

= dw z w

w f

i

n z f

n

n

1

)(

2

!

π

( ) ( )

∫ −= dw z ww f

i z f

π 2

1


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Drill

2. K$aluate the followin" line inte"ral in the

comple) plane by irect inte"ration

a- alon" a strai"ht line parallel to the 3 a)is

b- alon" the inicate path.

∫ +i

zdz

1

1

∫ C

dz z 2

& 272

(

%. #se (auchys theorem or inte"ral formula to e$aluate theinte"ral

∫ −C z

zdz

π 2

sin Where a- ( in the circle I/I 0 2 b- ( in the circle I/I 0 %


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!a:e ;ome Problems

2. K$aluate the followin" line inte"ral in the

comple) plane by irect inte"ration

a-

i- alon" the line y 0 x ii- alon" the inicate line

b- alon" the inicate path

( )∫ −C

dz z z 2

∫ C

dz z 2

21i

21i

272

721i


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A#RK+! SKRIKS

!heorem *II

aurents !heoremet C 2 an C % be two circles whose center at z o. et f z - be analytic in the re"ion R

between the circles. !hen f z - can be e)pane in a series of the form of

aurent Series

!he B series is calle the principle par% of the aurent series

convergen% in R.

( ) ( ) ( )( )

......2

0

2

0

12

02010 +−+

−++−+−+=

z z

b

z z

b z z a z z aa z f


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Definition

• If all the s are /ero, fz - is analytic at z 0 z &, an

we call /& a regular poin% .

• If n ≠ &, but all the s after n are /ero, fz - issai to ha$e a pole of or'er n at z 0 z &. If n 0 2 we

say that fz - has a simple pole.

• If there are an infinite number of s ifferent from

/ero, fz - has essensial singularity at z 0 z &.

• Coefficien% b1 of 2Cz G z &- is calle the residu of

fz - at z 0 z &.


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e)ample

( ) ( )11

332 +−

+ z z z

z

;as a pole of orer % at / 0 &, a pole of orer 8 at / 0 2,

an a simple pole at / 0 72


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e)ample

+++==∑=

z z z z n

z

z

e n

n

z

!2

111

! 2330

3

;as a pole of orer 8 at / 0 &, an resiu of e/C/8 is 2C%L


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Drill

2. For each of the followin" functions, say

whether the inicate point is regular , an

e$$en%ial $ingulari%y , or a pole, an if a

pole of what orer it is

a- Sin /-C/, / 0 &

b- cos /-C/8

, / 0 &c- /8 G 2-C/ G 2-8, / 0 2


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!;K RKSID#K !;KORKM


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Methos of Finin" Resiues

1. Laurent Series

2. Simple Pole

3. Multiple Poles


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A. Laurent Series

• K)ample1

)(−

= z e z f z

Solution

Fin Resiue R2- at / 0 2

( ) ( )

+−+−+

−=

−=

−

...2111

11.)(

21

z z z e

z ee z f z

...

1

++

−

= e z

e

!hus the resiue is coefficient of 2C/72- R2- 0 e


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J. Simple Pole

• K)ample( ) ( ) z z

z z f −+

=512

)(

Solution

Fin Resiue R7 - an R=-

Multiplyin" f/- by / G /&- ( ) ( ) ( ) z

z

z z

z z z f z

−=

−+

+=

+

525122

1)(

2

1

K$aluate the result at / 0 & R7 - 0 72C%%


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(. Multiple Poles

• K)ample( ) 3sin)(π −

= z

z z z f

Solution

Fin Resiue f/- at / 0

Multiply f/- by / G /&-m

m is inte"er "reater than or e6ual to the orer n of the pole-

Differentiate the result m G 2 times,

an e$aluate the resultin" at / 0 /&

i$ie by m72-L,

( ) z z z f z sin)(3 =−π

z z z z z dz

d sincos2sin

2

2

−=

z z z sincos2

−

1sincos2

−=− π π z


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!a:e ;ome Problems

•

Fin the resiues of the followin" functions at theinicate points. !ry to select the easiest metho.

( ) ( ) z z −+ 2231

at z 0 7%C8 an z 0 %

3

2

1 z

e iz

−

π

at z 0 e%πiC8

5cosh4

2

− z e z

at z 0 ln%

A li ti f R i !h


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( ) ( ) .inside of residuesof sum2 C z f idz z f C

⋅=∫ π ∫

=

= π θ 2

0 5

d I

Evaluation of Definite Integrals b !se of "#e $esi%ue "#eorem

"#e $esi%ue "#eorem

ere t#e integral aroun% C is in 'ounter'lo'(&ise

E)ample 1* +in% ∫ += π

θ θ

2

0 cos45d I

Application of Resiue !heorem

et< z = eiθ θ θ d iedz i=

,1dz iz d −−=θ 22

cos1−− +

=+

= z z ee ii θ θ

θ

( ) ( )∫ ∫ ∫ ++−=++−=

++−=

−

− π π π 2

0

2

0 2

2

0 1

1

212225

245

z z dz i

z z dz i

z z dz iz I


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!a:e ;ome Problems

+in%

∫ ∞

∞− +=

21 x

dx I

∫ ∞

+=

0 21

cos

x

dx x I

∫ ∞

∞−=

x

dx x I

sin

,1-

,2-

,3-


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Nisi7Nisi ui/

• Menentu:an Ja"ian Real u x ,y -Q an ba"ian Ima5iner

v x ,y -Q Fun"si Nomple:s

• Menentu:an apa:ah suatu fun"si :omple:s yan"

iberi:an termasu: fun"si analiti:

• Menentu:an apa:ah suatu fun"si :omple:s yan"

iberi:an termasu: fun"si harmoni:

• Menentu:an Resiu suatu fun"si :omple:s

• Menentu:an inte"ral "aris suatu fun"si :omple:s

• Menentu:an nilai inte"ral tertentu fun"si :omple:s

melalui !eorema Resiu

Mathematical Physics II-Function of Complex

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