a) Geometry often provides a helpful perspective on algebraic results, and this
ID: 3283498 • Letter: A
Question
a) Geometry often provides a helpful perspective on algebraic results, and this is certainly true in complex analysis. What algebraic results in complex analysis have useful geometric interpretations, and what insights into complex analysis do they afford?
b)Many integrals of real functions over the real number line can be evaluated using integration in the complex plane. Explain the theory behind this method, outline the general approach that should be taken, and comment on its uses and limitations.
Explanation / Answer
(A) Complex Analysis is so nice is because being holomorphic/analytic is an extremely strong condition.As opposed to real analysis, differentiability is a rather weak condition, so we have functions that are differentiable once but not twice etc. Real analysis is full of nasty counterexamples like the Weierstrass function which is continuous everywhere but differentiable nowhere.Analytic functions are C?C?, meaning they can be infinitely differentiated. Even more than that, analytic series is equal to its own Taylor series.
With regards to Algebraic Topology (AT), Hatcher does not focus much on the link between Complex Analysis and AT. Something interesting is that the Fundamental Theorem of Algebra can be proved in two different ways using Complex Analysis or Algebraic Topology
The reason complex analysis is nice is, to me, because it's about solutions to a certain very special kind of PDE called an "elliptic PDE": solutions to ?¯f=0?¯f=0, the Cauchy-Riemann equations, are automatically analytic by their very nature of being solutions to an elliptic PDE. Similarly, the integral formula arises and is useful (in different forms) in the study of similar PDE; the maximum principle is also a generally true fact for this sort of PDE, as is (one version of) the identity theorem
(B) in the he mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane
Contour integration is closely related to the calculus of residues,[4] a method of complex analysis.
One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods.
Contour integration methods include
direct integration of a complex-valued function along a curve in the complex plane (a contour)
application of the Cauchy integral formula
application of the residue theorem
One method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums
Applications of integral theorems
Applications of integral theorems are also often used to evaluate the contour integral along a contour, which means that the real-valued integral is calculated simultaneously along with calculating the contour integral.
Integral theorems such as the Cauchy integral formula or residue theorem are generally used in the following method:
a specific contour is chosen:
The contour is chosen so that the contour follows the part of the complex plane that describes the real-valued integral, and also encloses singularities of the integrand so application of the Cauchy integral formula or residue theorem is possible
application of Cauchy's integral theorem
The integral is reduced to only an integration around a small circle about each pole.
application of the Cauchy integral formula or residue theorem
Application of these integral formula gives us a value for the integral around the whole of the contour.
division of the contour into a contour along the real part and imaginary part
The whole of the contour can be divided into the contour that follows the part of the complex plane that describes the real-valued integral as chosen before (call it R), and the integral that crosses the complex plane (call it I). The integral over the whole of the contour is the sum of the integral over each of these contours.
demonstration that the integral that crosses the complex plane plays no part in the sum
If the integral I can be shown to be zero, or if the real-valued integral that is sought is improper, then if we demonstrate that the integral I as described above tends to 0, the integral along R will tend to the integral around the contour R + I.
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