Born: 21 Aug 1789 in Paris, France
Died: 23 May 1857 in Sceaux (near Paris), France
Born: 17 Sept 1826 in Breselenz, Hanover (now Germany)
Died: 20 July 1866 in Selasca, Italy
analysis is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of
physical problems. Contour integration, for example, provides a method of computing difficult integrals by investigating
the singularities of the function in regions of the complex plane near and between the limits of integration.
The most fundamental result of complex analysis is the Cauchy-Riemann equations, which give the conditions a
function must satisfy in order for a complex generalization of the derivative, the so-called complex derivative, to exist.
When the complex derivative is defined "everywhere," the function is said to be analytic. A single example of the
unexpected power of complex analysis is Picard's theorem, which states that an analytic function assumes every
complex number, with possibly one exception, infinitely often in any neighborhood of an essential singularity!
Here you will find at least 30 different Qualifying Exams in Complex Analysis from Temple University Since 1988. The
solutions to all problems are given here, too. I made sure that my solutions are detailed and correct. Most of them were
based on a course taught in the summer of 2003 by prof. Datskovsky. Also, they are ordered with respect to the chapter
that you might be reading. All relevant problems, in the comprehensive exams, from a certain chapter are solved under
the chapter's name so that you can study better. Please note that the probability that at least one of the solutions is
incorrect is approximately 1. So, be skeptical and please contact me with any corrections.
Study Guide for complex analysis: Notes Problems and Solutions
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Homework 1 and solutions (Cauchy's Integral Formula, uniform convergence of analytic functions, and the winding number)
Homework 2 and solutions (Singularities, Laurent Expansion, and construction of certain entire functions)
Homework 3 and solutions (Residue Calculus, evaluating infinite sums, and properties of meromorphic functions)
Homework 4 and solutions (Rouche's Theorem, and growth conditions of analytic functions)
Homework 5 and solutions (Schwarz' Lemma, Normal families, Riemann mapping theorem, infinite products and conformal maps)
16 Solved Problems from a first course in complex analysis
An elementary Semi-Proof of Picard's Little Theorem Using Conformal Mappings
The Legendre duplication formula for the Gamma Function
A formula for
Qualifying Exams on The WEB
Several Complex Variables
(1) Introductory Notes By R. Michael Range: Complex Analysis: A Brief Tour into Higher Dimensions
(2) An Online book on Several Complex Variables
(3) Some Homeworks and Solutions from R. Michael Range's Book:
"Holomorphic Functions and Integral Representations in Several Complex Variables":
Homework 1 (Holomorphic functions, Cauchy-Riemann Equations, Cauchy's integral formula for polydiscs, and power series)
Homework 2 (The derivative of a holomorphic function, implicit mapping theorem, Complex submanifolds, and biholomorphisms)
Homework 3 (Elementary extension phenomena: Extensions by Cauchy's integral formula, and Laurent series)
Homework 4 (Natural boundaries and pseudoconvexity: Domains of holomorphy, Hartogs and Levi pseudoconvexity)
Homework 5 (Holomorphic convexity, O(D)-Hulls, characterizations of domains of holomorphy, and complete Reinhardt domanis)
Homework 6 (Plurisubharmonic functions: Harmonic and subharmonic functions, submean value property, and pluriharmonic functions)
Extra 1. Let O be a bounded domain in R^N with C^2 boundary, and suppose that O is geometrically convex. Show that O is weakly convex.
Extra 2. Let O in R^{N} be a geometrically convex domain, p in ∂O, and q in O. Show that for all 0<t<1, we have (1-t)q+tp is in O.
Extra 3. Lewy's Example
Extra 4. Prove Brouwer's Fixed Point Theorem using Stokes' Theorem
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