Complex Analysis Flashcards
Three miracles of holomorphic functions?
- Contour integration: If f is holomorphic in Omega, then for appropriate closed paths in Omega the integral f(z)dz = 0.
- Regularity: If f is holomorphic, then f is indefinitely differentiable
- Analytic Continuation: If f and g are holomorphic functions in Omega which are equal in an arbitrarily small disc in Omega, then f = g everywhere in Omega.
Prove: | |z| - |w| | <= |z - w|
From triangle inequality. Stein 3
Define: holomorphic at the point z0, holomorphic on Omega, Entire
Examples?
Holomorphic if quotient [ f(z0 +h) - f(z0) ] / h converges to a limit when h –> 0. Here h in C, h != 0. Denote f’(z0) - derivative of f at z0.
- Polynomials = entire
- 1/z holomorphic on C - {0}
- Complex conjugation - not analytic
- Power series
Compare holomorphic functions to differentiable functions of one real variable
A holomorphic function satisfies much stronger properties than a differentiable function of one real variable.
- Holomorphic implies infinitely differentiable. Not true for real: f(x) = x^1/3 on R.
- Holomorphic implies analytic. Not true for real infinitely differentiable: f(x) = e^-1/x
Compare holomorphic functions to differentiable mappings R^2 –> R^2
Recall def of derivative for mapping of R^2 –> R^2.
Complex differentiation - derivative is complex number (rotation + dilation = 2 real degree of freedom)
Real derivatives - derivative is a matrix - Jacobian (4 real degrees of freedom)
Cauchy-Riemann Equations - link real and complex analysis
Complex valued function on open set of C holomorphic <=> Component functions are continuously differentiable and satisfy Cauchy-Riemann Equations
Note: We can weaken this to simply differentiable
Discuss the differential operators d/dz and d/dz bar
Stein 12
Discuss complex exponential vs complex geometric series
Stein 14
Discuss the radius of convergence of a power series. Prove existence
Stein 15. Converges absolutely on disk of convergence
Define trig functions on C. Euler formulas?
Power series definitions. e^iz + e^-iz / 2 = cos z etc.
Stein 16
Are power series holomorphic? Computation of derivative?
The power series f(z) = sum_n=0^infty a_n z^n defines a holomorphic function in its disc of convergence. The derivative of f is also a power series obtained by differentiating term by term the series for f, that is f’(z) = sum_n=0^infty n a_n z^n-1. Moreover, f’ has the same radius of convergence as f.
Use Hadamard + lim n^(1/n) = 1 for radius of convergence
Break into 3 pieces S_N, S’_N, and E_N
As a corollary: A power series is infinitely complex differentiable in its disc of convergence, everything obtained by termwise differentiation.
Stein 16
Define: analytic function
Analytic at z0 if exists a power series expansion centered at z0 with positive radius of convergence, if power series expansion at every point in domain, then called analytic
We will show: Holomorphic <=> Analytic
Define: parameterized curve, smooth, piecewise smooth, equivalent parameterizations, smooth curve, end points, closed, simple
A parameterized curve is a function z(t) which maps a closed interval [a,b] of R into the complex plane
Smooth if z’(t) exists and is continuous and z’(t) != 0 - i.e. an immersion
Piecewise smooth if z is continuous and if there exist points a = a0 < a1 < … < an = b where z(t) is smooth in the intervals [ak, ak+1]
Two parameterizations are equivalent if there exists a continuously differentiable bijection s -> t(s) from [c,d] to [a,b] s.t. t’(s) > 0 and z*(s) = z(t(s))
The smooth curve is the image in C. gamma- travels in opposite direction
closed if endpoints are equal z(a) = z(b). Simple if no self intersections
Standard parameterization of the circle?
z(t) = z_0 + re^it centered at z0 of radius r
Define: the integral of f along a curve gamma, length of gamma
Stein 21. Compare to Lee 291
How can we bound the size of a line integral
integral over gamma f(z) dz | <= sup |f(z)| * length(gamma)
Define: primitive of f
A function F that is holomorphic on Omega and st F’(z) = f(z) for all z in Omega
Prove: The Fundamental Theorem for Line Integrals - Complex setting
Corollary?
Lee 291, Stein 22, If f has a primitive F in Omega, and gamma is a curve in Omega that begins at w1 and ends at w2, then integral over gamma f(z) dz = F(w2) - F(w1).
If gamma is a closed curve in an open set omega and f is continuous and has a primitive in Omega, then integral over gamma of f(z) dz = 0.
Prove: f(z) = 1/z does not have a primitive in the open set C - {0}.
Stein 23. Just integrate it around a simple closed curve - circle - see that not 0. Can’t have primative.
Prove: If f is holomorphic in a region Omega and f’ = 0, then f is constant.
Stein 23 - 24
What is Goursat’s theorem? Proof
Thm. If Omega is an open set in C, and T < Omega a triangle whose interior is also contained in Omega, then
integral_T f(z) dz = 0
whenever f is holomorphic in Omega.
Proof. Create a nested sequence of triangles…stein 34-36
Prove: A holomorphic function in an open disk has a primitive in that disc
pg 37-39
Compare this to Lee 291-292 Fundamental Thm of Line Integrals - exact, closed, conservative
Prove: Cauchy’s theorem for a disk
How far can we generalize this approach?
If f is holomorphic in a disk, then integral_gamma f(z) dz = 0 for any closed curve gamma in that disk.
Proof. Simple application of existence of primitive pg 39
We can find a primitive in any contour that is simple enough to have a clear interior and simple enough s.t. we can construct polygonal paths in an open neighborhood of the contour and its interior
For the most general case, use Jordan curve theorem to show interior of any simple closed piecewise smooth curve is well-defined and simply connected - thus Cauchy holds
Show e^-pi x^2 is its own Fourier transform
pg 42-43
Discuss representation formulas
In particular integral representation formulas - Allow us to recover a function on a large set from its behavior on a smaller set
Examples from complex analysis? Harmonic functions?
