Tagged Questions

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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 ...
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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 ...
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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 ...
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 ...