1. 11. Complex Variable Theory
Complex Variables & Functions
Cauchy Reimann Conditions
Cauchy’s Integral Theorem
Cauchy’s Integral Formula
Laurent Expansion
Singularities
Calculus of Residues
Evaluation of Definite Inregrals
Evaluation of Sums
Miscellaneous Topics

2. Applications
Solutions to 2-D Laplace equation by means of conformal mapping.
Quantum mechanics.
Series expansions with analytic continuation.
Transformation between special functions, e.g., H (i x)  c K (x) .
Contour integrals :
Evaluate definite integrals & series.
Invert power series.
Form infinite products.
Asymptotic solutions.
Stability of oscillations.
Invert integral transforms.
Generalization of real quantities to describe dissipation, e.g.,
Refraction index: n  n + i k, Energy: E  E + i 

3. 1. Complex Variables & Functions
From § 1.8 :
(Ordered pair of real numbers )
Complex numbers :
Complex conjugate :
modulus
Polar representation :
argument 
Multi-valued function  single-valued in each branch
E.g., has m branches.
has an infinite number of branches.

4. 2. Cauchy Reimann Conditions
Derivative :
where limit is independent of path of  z  0.
Let    
 f  exists  & Cauchy- Reimann Conditions

5. f  exists  & Cauchy- Reimann Conditions
If the CRCs are satisfied,
 is independent of path of  z  0.
i.e., f  exists  CRCs satisfied.

6. Analytic Functions
f (z) is analytic in R  C  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  C \ {}.
z0 is a singular point of f (z) if f (z) doesn’t exist at z = z0 .

7. Example 11.2.1. z2 is Analytic  
f  exists & single-valued  finite z.
i.e., z2 is an entire function.

8. Example 11.2.2. z* is Not Analytic 
f  doesn’t exist  z, even though it is continuous every where.
i.e., z2 is nowhere analytic.

9. Harmonic Functions CRCs
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.
Let
 is analytic   
i.e., The real & imaginary parts of  must each satisfy a 2-D Laplace equation.
( u & v are harmonic functions )

10. CRCs
Contours of u & v are given by 
Thus, the slopes at each point of these contours are
CRCs  at the intersections of these 2 sets of contours
i.e., these 2 sets of contours are orthogonal to each other.
( u & v are complementary )

11. Derivatives of Analytic Functions
Let f (z) be analytic around z, then 
Proof :
f (z) analytic 
E.g. 
 Analytic functions can be defined by Taylor series of the
same coefficients as their real counterparts.

12. Example 11.2.3. Derivative of Logarithm
CRCs
for z within each branch.
Proof : 
 ln z is analytic within each branch. 
QED

13. Point at Infinity
Mathematica
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.

14. 3. Cauchy’s Integral Theorem
Contour = curve in z-plane
Contour integrals :
The t -integrals are just Reimann integrals
A contour is closed if
Closed contour integral :
( positive sense = counter-clockwise )

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

16. Example 11.3.1. zn on Circular Contour
on a circular contour 

17. Example 11.3.2. zn on Square Contour
Contour integral
from z = z0 to z = z1 along a straight line: 
Mathematica
For n = 1, each line segment integrates to i /2.
For other integer n, the segments cancel out in pairs. 

18. Proof of Theorem  Stokes theorem : S simply-connected
For S in x-y plane : 
QED
f analytic in S  CRCs
Note: The above (Cauchy’s) proof requires xu, etc, be continuous. Goursat’s proof doesn’t.

19. Multiply Connected Regions x
Value of integral is unchanged for any continuous deformation of C inside a region in which f is analytic.

20. 4. Cauchy’s Integral Formula
Let f be analytic in R & C  R. 
 z0 inside region bounded by C.
Cauchy’s Integral Formula y R
On C : z0 C
QED x

21. Example An Integral
C = CCW over unit circle centered at origin. y
(z+2)1 is analytic inside C.  x
z = 2
Alternatively 

22. Example 11.4.2. Integral with 2 Singular Factorsy
C = CCW over unit circle
centered at origin. x
z = 1/2
z = 1/2 

23. Derivatives
f analytic in R  C.
f (n) analytic inside C.

24. Example 11.4.3. Use of Derivative Formula
f analytic in R  C.
C = CCW over circle centered at a.
Let  

25. 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.
Proof :
 closed C   F  
So is F .
QED 
i.e., F is analytic in R.
Caution: this fails if R is multiply-connected (F multi-valued).

26. Further Applications If is analytic & bounded,
i.e., on a circle of radius r centered at the origin,
then Cauchy inequality
Proof :
C = circle of radius r .
Let 
QED
Corollary ( r   ) :
If f is analytic & bounded in entire z-plane, then f = const. ( Liouville’s theorem )

27. If f is analytic & bounded in entire z-plane, then f = const
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.
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 = 1 .
Repeat argument to the n  1 polynomial P / ( z  1 ) gives the next root z = 2 .
This can be repeated until P is reduced to a const, thus giving n roots.

28. 5. Laurent Expansion  Taylor series ( f analytic in R  C )
Mathematica
 Taylor series
( f analytic in R  C )
Let z1 be the closest singularity from z0 , then the radius of convergence is | z1 z0 |.
i.e., series converges for

29. Laurent Series Let f be analytic within an annular region  C1 : C2 :
Mathematica
C1 :
C2 : 

30. 
Laurent series
C within f ’s region of analyticity

31. Example 11.5.1. Laurent Expansion
Consider expansion about  f is analytic for
Expansion via binomial theorem :
Laurent series : 

32. 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.

33. Essential Singularities
e1/z is analytic except for z = 0.
 z = 0 is an essential singularity
sin z is analytic in the finite z-plane .
 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, ...

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

35. Example 11.6.2. Another Closed Loop
Consider
around the unit circle centered at z = 2.
Branch cut (x)-axis.
Mathematica
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.

36. Branch Point
For ,
Going around once any loop with z = 0 inside it results in a different f value.
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.
Branch point is a singularity (no f )
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.
By convention : f (x) is real in the principal branch.
Common choices of the principal branch are
&
A branch cut joins a branch point to another singularity, e.g., .

37. Example 11.6.3. ln z has an Infinite Number of Branches
Infinite number of branches
 z = 0 is the branch point (of order ).
Similarly for the inverse trigonometric functions.
p = integers   z p is single-valued.
p = rational = k / m  z p is m-valued.
p = irrational  z p is -valued.

38. Example 11.6.4. Multiple Branch Points
 2 branch points at z = 1.
 1 simple pole at z = .
Mathematica
Let 
Let the branch cuts for both ( z  1 )1/2 be along the (x )-axis, i.e.,
in the principal branch.

39. Analytic Continuation
f (z) is analytic in R  C  f has unique Taylor expansion at any z0  R .
Radius of convergence is distance from z0 to nearest singularity z1 .
Coefficients of Taylor expansion  f (n) (z0) .
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).

40. 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

41. Example 11.6.5. Analytic Continuation
Consider
For any point P on the line sement,

42.  f1 & f2 are expansions of the same function 1/z.
Analytic continuation can be carried out for functions expressed in forms other than series expansions.
E.g., Integral representations.