Mathematical Physics II-Function of Complex
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Transcript of 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