11. Complex Variable Theory

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(ix)  cK (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 + ik, 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 fdepend 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 & vare harmonic functions )

10. CRCs Contours of u & vare 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 & vare 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 zwithin 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. znon Circular Contour  on a circular contour 

17. Example 11.3.2. znon Square Contour Contour integral from z = z0to z = z1along 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 y R C C R C C  x Value of integral is unchanged for any continuous deformation of Cinside 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 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 

22. 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 

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 connectedR &  closed C  R, then f (z) is analytic throughout R.  closed C   F  Proof :  So isF . 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. ( 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 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  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 z0except for the point z0.  Laurent series about z0exists. If the lowest power of zz0 in the series is n, then z0is 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/zis analytic except for z = 0. z = 0 is an essential singularity sin zis analytic in the finite z-plane. t = 0 orz =  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 : fis 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 itresults 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 fin 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 n = 0, 1, 2, ...  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 z0to 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 & grepresent 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 = 1is indeed a branch point ) Path encircling z = +1: f (z) double-valued. ( z = +1is 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 & f2are 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.