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

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 

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.

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

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

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 .

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

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.

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 )

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 )

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.

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

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.

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 )

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

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

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. 

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.

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

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

Example 11.4.1. An Integral C = CCW over unit circle centered at origin. y (z+2)1 is analytic inside C.  x z = 2 Alternatively 

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

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

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

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).

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 )

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.

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

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

 Laurent series C within f ’s region of analyticity

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

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.

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, ...

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.

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.

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., .

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.

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.

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).

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

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

 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.