This document discusses complex variables and functions. It covers topics such as: - Cauchy-Riemann conditions, which must be satisfied for a complex function to be analytic/differentiable - Cauchy's integral theorem, which states that the integral of an analytic function around a closed contour is zero - Harmonic functions, whose real and imaginary parts satisfy the 2D Laplace equation - Examples of calculating contour integrals of functions like zn along circular and square paths The document provides proofs of theorems like Cauchy's integral theorem using techniques like Stokes' theorem. It also discusses simply and multiply connected regions in relation to contour integrals.

1. LD COLLEGE OF
ENGINEERING

2. 11. Complex Variable Theory
1. Complex Variables & Functions
2. Cauchy Reimann Conditions
3. Cauchy’s Integral Theorem
4. Cauchy’s Integral Formula
5. Laurent Expansion
6. Singularities
7. Calculus of Residues
8. Evaluation of Definite Inregrals
9. Evaluation of Sums
10. Miscellaneous Topics

3. Applications
1. Solutions to 2-D Laplace equation by means of conformal mapping.
2. Quantum mechanics.
3. Series expansions with analytic continuation.
4. Transformation between special functions, e.g., Hν (i x) ↔ c Kν (x) .
5. Contour integrals :
a) Evaluate definite integrals & series.
b) Invert power series.
c) Form infinite products.
d) Asymptotic solutions.
e) Stability of oscillations.
f) Invert integral transforms.
6. Generalization of real quantities to describe dissipation, e.g.,
Refraction index: n → n + i k, Energy: E → E + i Γ

4. 1. Complex Variables & Functions
{ }; ,z x i y x y= = + ∀ ∈£ ¡Complex numbers :
(Ordered pair of
real numbers )
Complex conjugate : *z x i y= −
Polar representation :
i
z r e θ
=
2 2
r x y= + modulus
1
tan
y
x
θ −
= argument
→ cos sini
e iθ
θ θ= +
From § 1.8 :
Multi-valued function → single-valued in each branch
E.g., has m branches.
has an infinite number of branches.
( )2 /1/ 1/ i n mm m
z r e
θ π+
=
( )ln ln 2z r i nθ π= + +
( )2i n
r e
θ π+
=

5. 2. Cauchy Reimann Conditions
( )
( )
d f z
f z
d z
′=Derivative :
( )
0
lim
z
f z
zδ
δ
δ→
=
( ) ( )
0
lim
z
f z z f z
zδ
δ
δ→
+ −
=
where limit is independent of path of δ z → 0.
Let ( ) ( ) ( )f z u z i v z= +
f u i vδ δ δ= +z x i yδ δ δ= +→
∴ f ′ exists → & Cauchy- Reimann
Conditions
f u i v
z x i y
δ δ δ
δ δ δ
+
=
+
→
z xδ δ= →
0 0
lim lim
z x
f u v
i
z x xδ δ
δ δ δ
δ δ δ→ →
 
= + ÷
 
u v
i
x x
∂ ∂
= +
∂ ∂
z yδ δ= →
0 0
lim lim
z y
f u v
i
z y yδ δ
δ δ δ
δ δ δ→ →
 
= − + ÷
 
u v
i
y y
∂ ∂
= − +
∂ ∂
u v
x y
∂ ∂
=
∂ ∂
u v
y x
∂ ∂
= −
∂ ∂
z x i y= +

6. → is independent of path of δ z → 0.
f ′ exists → & Cauchy- Reimann
Conditions
u v
x y
∂ ∂
=
∂ ∂
u v
y x
∂ ∂
= −
∂ ∂
If the CRCs are satisfied,
( ) ( ) ( )f z u z i v z= +z x i y= +
( ) ( ),f z f x y= →
f f
f x y
x y
δ δ δ
∂ ∂
= +
∂ ∂
u v u v
i x i y
x x y y
δ δ
   ∂ ∂ ∂ ∂
= + + + ÷  ÷∂ ∂ ∂ ∂   
( )
u v
f i x i y
x x
δ δ δ
 ∂ ∂
= + + ÷∂ ∂ 
f u v
i
z x x
δ
δ
∂ ∂
= +
∂ ∂
i.e., f ′ exists ↔ CRCs satisfied.

7. Analytic Functions
f (z) is analytic in R ⊆  ↔ f ′ exists & single-valued in R.
Note: Multi-valued functions can be analytic within each branch.
f (z) is an entire function if it is analytic ∀ z ∈  {∞}.
z0 is a singular point of f (z) if f ′(z) doesn’t exist at z = z0 .

8. Example 11.2.1. z2
is Analytic
( ) 2
f z z=
z x i y= +
2 2
2x y i x y= − + u i v= +
→
2 2
2
u x y
v x y
= −
=
2
u
x
x
∂
=
∂
→
v
y
∂
=
∂
2
u
y
y
∂
= −
∂
v
x
∂
= −
∂
∴ f ′ exists & single-valued ∀ finite z.
i.e., z2
is an entire function.

9. Example 11.2.2. z* is Not Analytic z x i y= +
( ) *f z z x i y= = − u iv= +
→
u x
v y
=
= −
1
u
x
∂
=
∂
→ 1
v
y
∂
≠ − =
∂
0
u
y
∂
=
∂
v
x
∂
= −
∂
∴ f ′ doesn’t exist ∀ z, even though it is continuous every where.
i.e., z2
is nowhere analytic.

10. Harmonic Functions
By definition, derivatives of a real function f depend only on the local behavior of f.
But derivatives of a complex function f depend on the global behavior of f.
u v
x y
∂ ∂
=
∂ ∂
u v
y x
∂ ∂
= −
∂ ∂
CRCs
Let ( )z u ivψ = +
ψ is analytic →
u v
x y
∂ ∂
=
∂ ∂
u v
y x
∂ ∂
= −
∂ ∂
∴
2 2
2
u v
x x y
∂ ∂
=
∂ ∂ ∂
2
v
y x
∂
=
∂ ∂
2
2
u
y
∂
= −
∂
2 2
2
v u
y y x
∂ ∂
=
∂ ∂ ∂
2
u
x y
∂
=
∂ ∂
2
2
v
x
∂
= −
∂
→
2 2
2 2
0
u u
x y
∂ ∂
+ =
∂ ∂
2 2
2 2
0
v v
x y
∂ ∂
+ =
∂ ∂
i.e., The real & imaginary parts of ψ must each satisfy a 2-D Laplace equation.
( u & v are harmonic functions )

11. u v
x y
∂ ∂
=
∂ ∂
u v
y x
∂ ∂
= −
∂ ∂
CRCs
Contours of u & v are given by ( ),u x y c=
→ 0
u u
du d x d y
x y
∂ ∂
= + =
∂ ∂
( ),v x y c′=
0
v v
dv d x d y
x y
∂ ∂
= + =
∂ ∂
i.e., these 2 sets of contours are orthogonal to each other.
( u & v are complementary )
u
u
u
d y x
m
ud x
y
∂
  ∂
= = − ÷ ∂ 
∂
Thus, the slopes at each point of these contours are
v
v
v
d y x
m
vd x
y
∂
  ∂
= = − ÷ ∂ 
∂
CRCs → at the intersections of these 2 sets of contours1u vm m = −

12. Derivatives of Analytic Functions
( )
( )
d f x
g x
d x
=
Let f (z) be analytic around z, then
→
( )
( )
d f z
g z
d z
=
Proof :
f (z) analytic → ( )
( )f x i y
f z
x
∂ +
′ =
∂
( )
x z
d f x
d x =
= ( )g z=
z x i y= +
E.g.
1
n
nd x
n x
d x
−
= →
1
n
nd z
n z
d z
−
=
∴ Analytic functions can be defined by Taylor series of the
same coefficients as their real counterparts.

13. Example 11.2.3.
Derivative of Logarithm
ln 1d z
d z z
=
Proof : ( )ln ln 2z r i nθ π= + +
for z within each branch.
u iv= + →
ln
2
u r
v nθ π
=
= +
u v
x y
∂ ∂
=
∂ ∂
u v
y x
∂ ∂
= −
∂ ∂
CRCs
1u r
x r x
∂ ∂
=
∂ ∂ 2
x
r
=
2 2
r x y= +
1
tan
y
x
θ −
=
v
y
∂
=
∂
12
2
1
1
y
y x x
θ
−
 ∂
= + ÷∂  
2
x
r
=
1u r
y r y
∂ ∂
=
∂ ∂ 2
y
r
=
v
x
∂
= −
∂
12
2 2
1
y y
x x x
θ
−
 ∂  
= + − ÷ ÷∂   
2
y
r
= −
→ ln z is analytic within each branch.
∴
ln lnd z z
d z x
∂
=
∂
u v
i
x x
∂ ∂
= +
∂ ∂ 2 2
x y
i
r r
= −
1
x i y
=
+
1
z
= 2
*r z z=
QED

14. Point at Infinity
The entire z-plane can be mapped 1-1 onto the surface of the unit sphere,
the north (upper) pole of which then represents all points at infinity.
Mathematica

15. 3. Cauchy’s Integral Theorem
Contour integrals :
( ) ( ) ( ):C z t x t i y t= +Contour = curve in z-plane
( ) ( ) ( ) ( ) ( )
1
0
t
C
t
dz f z dt x t i y t u t i v t= + +      ∫ ∫
( ) ( ) ( )f z u z i v z= +
( )
1
0
t
t
dt xu yv i xv yu= − + +  ∫
The t -integrals are just
Reimann integrals
Closed contour integral :
( ) ( )0 1z t z t=A contour is closed if
( positive sense = counter-clockwise )( )
C
f zd z
∫Ñ

16. Statement of Theorem
Let C be a closed contour inside a simply connected region R ⊆ .
If f (z) is analytic in R, then
( ) 0
C
f zd z =
∫Ñ
A region is simply connected if every closed curve in it
can be shrunk continuously to a point.
Cauchy’s intgeral theorem

17. Example 11.3.1. zn
on Circular Contour
( )
2
11
0
i nn n
C
z i r d ed z
π
θ
θ ++
=
∫∫Ñ
i
z r e θ
= →
i
d z i r e dθ
θ= on a circular contour
∴
( )
1 2
1
0
2
0
integers & 1
1
1
n
i nr
e n n
n
i d n
π
θ
π
θ
+
+
= ≠ − +
= 
 = −


∫
0 integers & 1
2 1
n n
i nπ
= ≠ −
= 
= −

18. Example 11.3.2. zn
on Square Contour
Contour integral
from z = z0 to z = z1 along a straight line:
( ) ( )0 1 0 : 0 1z t z z z t t= + − →
( )1 0d z z z dt= −→
( ) ( ) ( )
1
0
1
1 0 0 1 0
0
z
z
d z f z dt z z f z z z t= − + −  ∫ ∫
→
2 1
0 integers & 1
n
C
i n
zd z
n n
π = −
= 
= ≠ −∫Ñ
Mathematica
For n = −1, each line segment integrates to iπ /2.
For other integer n, the segments cancel out in pairs.

19. f analytic in S → CRCs
Proof of Theorem ( ) 0
C
f zd z =
∫Ñ
z x i y= +
f u i v= +
→ ( ) ( ) ( )
C C C
f z id z u d x v d y v d x u d y= +− +
∫ ∫ ∫Ñ Ñ Ñ
Stokes
theorem :
S S
dd
∂
= ×∇××
∫ ∫ σVr V
Ñ
For S in x-y plane : ( )x y
y x
S S
V V
d x d yV d x V d y
x y
∂
∂ ∂
= −+  ÷
∂ ∂ ∫ ∫Ñ
→ ( )
C S S
v u u v
f z d x d y i d x d yd z
x y x y
   ∂ ∂ ∂ ∂
= − − + − ÷  ÷
∂ ∂ ∂ ∂   ∫ ∫ ∫Ñ
0=
S simply-connected
QED
u v
x y
∂ ∂
=
∂ ∂
u v
y x
∂ ∂
= −
∂ ∂
Note: The above (Cauchy’s) proof
requires ∂xu, etc, be continuous.
Goursat’s proof doesn’t.

20. Multiply Connected Regions
y
x
R
R′
C
C
( ) 0
CC C C
d z f z
 
 + − + =
 
 
∫∫∫∫
C
C
( ) 0
C C
d z f z
 
 + =
 
 
∫∫
→ ( ) ( )
C C
d z f z d z f z=
∫∫
Value of integral is unchanged for any continuous
deformation of C inside a region in which f is analytic.

21. 4. Cauchy’s Integral Formula
Let f be analytic in R & C ⊂ R.
→ ( )
( )
0
0
1
2
C
f z
f z d z
i z zπ
=
−∫Ñ
∀ z0 inside region bounded by C.
Cauchy’s Integral Formula
R
C
y
x
z0
( ) ( )
0 0
CC
f z f z
d z d z
z z z z
=
− −∫∫ ÑÑ
( ) ( )
2
0
0 0
0
lim
C
i
i
r
f z f z
d i r ed z
z z r e
π
θ
θ
θ
→
=
− ∫∫Ñ
On C : 0
i
z z r e θ
− = i
d z i r e dθ
θ=
( )02 i f zπ= QED

22. Example 11.4.1. An Integral
( )
1
2
C
I d z
z z
=
+∫Ñ C = CCW over unit circle centered at origin.
( )
1 1 1 1
2 2 2z z z z
 
= − ÷
+ + 
→
iπ=
0
1
2
2 z
I i
z
π
=
=
+
Alternatively
( )
( )
0
0
1
2
C
f z
f z d z
i z zπ
=
−∫Ñ
(z+2)−1
is analytic inside C.
1 1 1
2 2
C C
I d z d z
z z
 
 = −
 +
 
∫ ∫Ñ Ñ ( )
1
2 0
2
iπ= − iπ=
y
x
→
z = −2

23. Example 11.4.2.
Integral with 2 Singular Factors
2
1
4 1
C
I d z
z
=
−∫Ñ C = CCW over unit circle
centered at origin.
( ) ( )2
1 1
4 1 2 1 2 1z z z
=
− + −
1 1 1
2 2 1 2 1z z
 
= − ÷
− + 
1 1 1
4 1/ 2 1/ 2z z
 
= − ÷
− + 
∴ ( )
1
2 1 1
4
I iπ= − 0=
( )
( )
0
0
1
2
C
f z
f z d z
i z zπ
=
−∫Ñ
y
x
z = −1/2 z = 1/2

24. Derivatives
( )
( )
0
0
1
2
C
f z
f z d z
i z zπ
=
−∫Ñ
( )
( )
0
0 0
1
2
C
f z
f z d z
i z z zπ
 ∂
′ =  ∂ − ∫Ñ
( )
( )
2
0
1
2
C
f z
d z
i z zπ
=
−∫Ñ
( )
( )
( )
( )
0 1
0
!
2
n
n
C
f zn
f z d z
i z zπ +
=
−∫Ñ
( )
( )
( )
0 3
0
2
2
C
f z
f z d z
i z zπ
′′ =
−∫Ñ
f analytic in R ⊃ C.
f (n)
analytic inside C.

25. Let →
Example 11.4.3.
Use of Derivative Formula
( )
2
4
sin
C
z
I d z
z a
=
−∫Ñ C = CCW over circle centered at a.
( )
( )
( )
( )
0 1
0
!
2
n
n
C
f zn
f z d z
i z zπ +
=
−∫Ñ
f analytic in R ⊃ C.
( )
( )
( )
2
3
4
3! sin
2
C
z
f a d z
i z aπ
=
−∫Ñ( ) 2
sinf z z=
( ) 2sin cosf z z z′ =
( ) ( )2 2
2 cos sinf z z z′′ = −
( )
( )3
8sin cosf z z z= −
→ ( )
( )32
3!
i
I f a
π
=
8
sin cos
3
i
a a
π
= −

26. Morera’s Theorem
Morera’s theorem :
If f (z) is continuous in a simply connected R &
∀ closed C ⊂ R,
then f (z) is analytic throughout R.
( ) 0
C
f zd z =
∫Ñ
( ) 0
C
f zd z =
∫ÑProof : ∀ closed C → ∃ F ∋ ( ) ( ) ( )
2
1
2 1
z
z
F z F z d z f z− =
∫
∴
( ) ( )
( ) ( ) ( )
2
1
2 1
1 1
2 1 2 1
1
z
z
F z F z
f z d z f z f z
z z z z
−
− = −  − − ∫
( ) ( )
2
1
1 1
2 1
1
z
z
d z z z f z
z z
′≈ − +  − ∫ L ( ) ( )2 1 1
1
2
z z f z′≈ − +L
→ ( )
( ) ( )
2 1
2 1
1
2 1
lim
z z
F z F z
f z
z z→
−
=
−
( )1F z′= i.e., F is analytic in R. So is F ′.
QED
Caution: this fails if R is multiply-connected (F multi-valued).

27. If is analytic & bounded,
i.e., on a circle of radius r centered at the origin,
then Cauchy inequality
Further Applications
( ) n
n
n
f z a z= ∑
( )f z M≤
n
na r M≤
Proof :
( )
( )
1
0
!
n
na f
n
=
( )
1
1
2 n
C
f z
d z
i zπ +
=
∫Ñ
( )
( )
( )
( )
0 1
0
!
2
n
n
C
f zn
f z d z
i z zπ +
=
−∫Ñ
( )
1
1
2
2 n
M r
r
r
π
π +
 
≤  
 
Let →( ) ( )max
z r
M r f z
=
=
( )
1
1
2
n n
z r
f z
a d z
zπ +
=
=
∫Ñ
QED
C = circle of radius r .
Corollary ( r → ∞ ) :
If f is analytic & bounded in entire z-plane, then f = const. ( Liouville’s theorem )

28. If f is analytic & bounded in entire z-plane, then f = const. ( Liouville’s theorem )
→ If f is analytic & non-constant, then ∃ at least one singularity in the z-plane.
E.g., f (z) = z is analytic in the finite z-plane, but has singularity at infinity.
So is any entire function.
Fundamental theorem of algebra :
Any polynomial with n > 0 & a ≠ 0 has n roots.( )
0
n
k
k
k
P z a z
=
= ∑
Proof :
If P has no root, then 1/P is analytic & bounded ∀ z.
→ P = const. ( contradiction )
∴ P has at least 1 root, say at z = λ1 .
Repeat argument to the n − 1 polynomial P / ( z − λ1 ) gives the next root z = λ2 .
This can be repeated until P is reduced to a const, thus giving n roots.

29. → Taylor series
( f analytic in R ⊃ C )
5. Laurent Expansion
( )
( )
0
0
1
2
C
f z
f z d z
i z zπ
=
−∫Ñ( )
( )1
2
C
f z
f z d z
i z zπ
′
′=
′−∫Ñ
( )0 0
1 1
z z z z z z
=
′ ′− − − −
1
0
0 0
1
1
z z
z z z z
−
 −
= − ÷′ ′− − 
0
00 0
1
n
n
z z
z z z z
∞
=
 −
=  ÷′ ′− − 
∑
( ) ( )
( )
( )
0 1
0
1
2
n
n
n
C
f z
f z z z d z
i z zπ
∞
+
=
′
′= −
′−
∑ ∫Ñ
( )
( )
( )
( )
0 1
0
!
2
n
n
C
f zn
f z d z
i z zπ +
=
−∫Ñ
( )
( ) ( )
( )0
0
0 !
n
n
n
z z
f z f z
n
∞
=
−
= ∑
Let z1 be the closest singularity from z0 , then the radius of convergence is | z1− z0 |.
i.e., series converges for 0 1 0z z z z− < −
Mathematica

30. Laurent Series
Mathematica
Let f be analytic within an annular region
0r z z R≤ − ≤
→
( )
( )
0
0
1
2
C
f z
f z d z
i z zπ
=
−∫Ñ
( )
( )
( )
( )
0 1
0
!
2
n
n
C
f zn
f z d z
i z zπ +
=
−∫Ñ
( )
( )
21
1
2
CC
f z
f z d z
i z zπ
 
′  ′= −
  ′−
  
∫∫Ñ Ñ
( ) ( )
( )
( ) ( )
( ) ( )
21
0 01 1
0 00 0
1 1 1
2 2
C
n n
n n
n n
C
f z
f z z z z z f zd z d z
i iz z z zπ π
∞ ∞
+ +
= =
′
′ ′ ′ ′= − + −
′− −
∑ ∑ ∫∫ ÑÑ
( )
0
00 0 0 0
1 1 1
n
n
z z
z z z z z z z z z z
∞
=
 −
= =  ÷′ ′ ′ ′− − − − − − 
∑C1 :
0
00 0
1 1
n
n
z z
z z z z z z
∞
=
 ′−
= −  ÷′− − − 
∑C2 :
→

31. ( ) ( )
( )
( ) ( )
( ) ( )
21
0 01 1
0 00 0
1 1 1
2 2
C
n n
n n
n n
C
f z
f z z z z z f zd z d z
i iz z z zπ π
∞ ∞
+ +
= =
′
′ ′ ′ ′= − + −
′− −
∑ ∑ ∫∫ ÑÑ
( )
( ) ( ) ( )
( )
( )2 2
1
0 01 1
0 0 0
1
C C
n n
n n
n n
f z
z z f z z zd z d z
z z z z
∞ −
+ +
= = −∞
′
′ ′′ ′− = −
′− −
∑ ∑∫ ∫Ñ Ñ
→ ( ) ( )0
n
n
n
f z a z z
∞
= −∞
= −∑ Laurent series
( )
( )
1
0
1
2
n n
C
f z
a d z
i z zπ +
′
′=
′ −∫Ñ C within f ’s region of analyticity

32. Consider expansion about → f is analytic for
Example 11.5.1.
Laurent Expansion
( ) ( )0
n
n
n
f z a z z
∞
= −∞
= −∑
( )
( )
1
0
1
2
n n
C
f z
a d z
i z zπ +
′
′=
′−∫Ñ
( )
( )
1
1
f z
z z
=
−
0 0z = 0 1z< <
( )
1 1
1
f z
z z
 
= − + ÷
− 
Expansion via binomial theorem :
0
1 n
n
z
z
∞
=
 
= − + ÷
 
∑
Laurent series :
( )1
1 1
2 1
n n
C
a d z
i z z zπ +
=
−∫Ñ
1 1
0
n
otherwise
− ≥ −
= 

2
0
1
2
k
n
k
C
z
d z
i zπ
∞
+
=
= − ∑ ∫Ñ
→ ( )
1
n
n
f z z
∞
= −
= − ∑

33. 6. Singularities
Poles :
Point z0 is an isolated singular point if f (z) is analytic
in a neighborhood of z0 except for the point z0 .
→ Laurent series about z0 exists.
If the lowest power of z− z0 in the series is −n,
then z0 is called a pole of order n.
Pole of order 1 is called a simple pole.
Pole of order infinity is called an essential singularity.

34. Essential Singularities
1/
0
1
!
z n
n
e z
n
∞
−
=
= ∑
0
1
!
n
n
z
n= −∞
= ∑ → z = 0 is an essential singularity
e1/z
is analytic except for z = 0.
sin z is analytic in the finite z-plane .
( )
( )
2 1
0
sin
2 1 !
n
n
n
z z
n
∞
+
=
−
=
+
∑
( )
( )
1
2 1
0
lim sin lim
2 1 !
n
n
z t
n
z t
n
−
− +
→ ∞ →
= −∞
−
=
+
∑ → t = 0 or z = ∞
is an essential singularity
A function that is analytic in the finite z-plane except for poles is meromorphic.
E.g., ratio of 2 polynomials, tan z, cot z, ...
A function that is analytic in the finite z-plane is an entire function.
E.g., ez
, sin z, cos z, ...

35. Starting at θA = 0, we have
Consider
around the unit circle centered at z = 0.
Example 11.6.1. Value of z1/2
on a Closed Loop
( ) 1/2
f z z=
2 values at each point : f is double-valued
( ) /2i
f z e θ
=
Mathematica
Branch cut
(+x)-axis.
i
z e θ
=
Value of f jumps when branch cut is crossed.
∴ Value of f jumps going around loop once.

36. Example 11.6.2. Another Closed Loop
Mathematica
Branch cut
(−x)-axis.
Consider
around the unit circle centered at z = 2.
2 i
z e θ
= +
( ) 1/2
f z z=
( ) / 2i
f z r e φ
=
( )2
10 6cosr π θ= − −
sin
tan
2 cos
θ
φ
θ
=
+
No branch cut is crossed going around loop.
→ No discontinuity in value of f .
If branch cut is taken as (+x)-axis, f jumps
twice going around loop & returns to the
same value.

37. Branch PointFor ,
1.Going around once any loop with z = 0 inside it results in a different f value.
2.Going around once any loop with z = 0 outside it results in the same f value.
→ Any branch cut must start at z = 0.
z = 0 is called the branch point of f.
( ) 1/2
f z z=
The number of distinct branches is called the order of the branch point.
The default branch is called the principal branch of f.
Values of f in the principal branch are called its principal values.
Common choices of the principal branch are
0 2θ π= → & θ π π= − →
Branch point is a
singularity (no f ′)
By convention : f (x) is real in the principal branch.
A branch cut joins a branch point to another singularity, e.g., ∞.

38. Example 11.6.3.
ln z has an Infinite Number of Branches
i
z r e θ
= ( )2i n
r e
θ π+
= n = 0, ±1, ±2, ...
→ ( )ln ln 2z r i nθ π= + + Infinite number of branches
→ z = 0 is the branch point (of order ∞).
ln 1d z
d z z
=
Similarly for the inverse trigonometric functions.
( )exp lnp
z p z= ( ) ( )exp ln exp 2p r i i pnθ π= +  
p = integers → → z p
is single-valued.( )exp 2 1i pnπ =
p = rational = k / m → z p
is m-valued.
p = irrational → z p
is ∞-valued.

39. Let →
Example 11.6.4. Multiple Branch Points
( ) ( )
1/22
1f z z= − ( ) ( )
1/2 1/2
1 1z z= + − → 2 branch points at z = ±1.
( )
1/2
1/22
20
1
lim 1 lim 1
z t
z
t→ ∞ →
 
− = − ÷
 
( )
1/22
0
1
lim 1
t
t
t→
= − ( )1/2 2
0
0
1
lim
k
k
t
k
C t
t
∞
→
=
= −∑
3
0
1 1 1
lim
2 8t
t t
t→
 
= − − + ÷
 
L → 1 simple pole at z = ∞.
1
1
i
i
z s e
z t e
φ
ϕ
+ =
− =
( ) ( ) / 2i
f z st e
φ ϕ+
=
Let the branch cuts for both ( z ± 1 )1/2
be along the (−x )-axis, i.e.,
in the principal branch.
,φ ϕ π π= − →
Mathematica

40. Analytic Continuation
f (z) is analytic in R ⊂  → f has unique Taylor expansion at any z0 ∈ R .
Radius of convergence is distance from z0 to nearest singularity z1 .
1. Coefficients of Taylor expansion ∝ f (n)
(z0) .
2. f (n)
(z0) are independent of direction.
→ f (z) known on any curve segment through z0
is enough to determine f (n)
(z0) ∀ n.
∴ Let f (z) & g (z) be analytic in regions R & S, respectively.
If f (z) = g (z) on any finite curve segment in R  S,
then f & g represent the same analytic function in R  S.
f ( or g ) is called the analytic continuation of g ( f ) into R (S).

41. Path encircling both branch points:
f (z) single-valued.
( effectively, no branch line crossed )
Path in between BPs:
f (z) has 2 branches.
( effective branch line = line joining BPs. )
Path encircling z = −1 :
f (z) double-valued.
( z = −1 is indeed a branch point )
Path encircling z = +1 :
f (z) double-valued.
( z = +1 is indeed a branch point )Hatched curves
= 2nd
branch
curve = path
( )
( )
( )
1/
1/2
1 2
2
2
/
1
1
1z
z
z
−
−
+

42. Example 11.6.5. Analytic Continuation
Consider ( ) ( ) ( )1
0
1
n n
n
f z z
∞
=
= − −∑
( ) ( )1
2
0
nn
n
f z i z i
∞
−
=
= −∑
1 1z∀ − <
1z i∀ − <
For any point P on the line sement,
( )1 0 1P r i r= + ≤ ≤
( ) ( ) ( )1
0
1
n n
n
f P r ri
∞
=
= − − +∑
( ) ( )1
2
0
1
nn
n
f P i r r i
∞
−
=
= + −  ∑

43. ( ) ( ) ( )1
0
1
n n
n
f P r ri
∞
=
= − − +∑ ( ) ( )1
2
0
1
nn
n
f P i r r i
∞
−
=
= + −  ∑
( )
( )1
1
1 1
f P
r ri
=
+ − +
( ) ( )1
2
0
1
n
n
f P i ir r
∞
−
=
= − −  ∑
( )
1
1r i
=
+
( )
1 1
1 1
i
ir r
−
=
− − −  
( )
1 1
1
i
r i
−
=
− ( )
1
1r i
=
+
( )1f P=
∴ f1 & f2 are expansions of the same function 1/z.
1
P
=
( )1P r i= +
Analytic continuation can be carried out for functions
expressed in forms other than series expansions.
E.g., Integral representations.