7
votes
1answer
137 views

Proving that a function is analytic

I'm struggling with the following problem: Problem: Suppose that $h$ is a continuous function on a simple closed curve $\gamma$. Define $$ H(w) = \oint_{\gamma} \frac{h(z)}{z - w} \, dz. $$ Show ...
1
vote
1answer
42 views

Integral Continuation $\Gamma(z)=\int_{0}^{1} e^{-t} t^{z-1} dt +\int_{1}^{\infty} e^{-t} t^{z-1}dt$

I am trying to obtain an analytical continuation for $\Gamma(z)$ into the region of the complex plane characterized by $\Re(z) \leq 0$ but am stuck. Starting from the integral definition of ...
1
vote
1answer
42 views

Integrating the function Im(z) on a variety of contours.

I've been asked to evaluate $\int_C Im(z) dz$ for a variety of contours, which I've had no issue in doing. For the sake of clarity, these contours included the upper and lower halves of the circle ...
1
vote
1answer
101 views

Analytic continuation of function given as integral

I have a function $I(D)$ defined by the following integral representation $$ I(D)=\int_0^\infty\mathrm{d}\alpha\,(1+2\alpha)^{-D/2} $$ which is clearly only sensible for $D>2$. The result of the ...
2
votes
1answer
282 views

Area and locally one-to-one analytic mappings of the unit disk.

We learned about conformal mappings and various properties of locally one-to-one, analytic mappings of the unit disk. I am having trouble with the following problem, can anyone help? Let $f(z) ...
6
votes
3answers
375 views

Expressing the area of the image of a holomorphic function by the coefficients of its expansion

I have the following problem. Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ denote the Lebesgue measure of a set ...