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I want to calculate the following complex integrals of the following functions:

(.) $f(z)$= $\frac{e^{senz}}{|z|^3}$ over the circle {$z$/ |$z$|=$2$}

(..)$\int_{|z|=\frac{1}{2}} \sqrt{z^2-1}\,dz$ where the root is defined taking the branch $(0,2\pi)$

Im pretty sure these two integral can be solved by using Cauchy theorem so for the first one (.) since $e^{senz}$ is analytic over the whole complex plane but $\frac{e^{senz}}{|z|^3}$ is analytical with continuos derivative all over the complex plane without the points where $|z|^3$ = $0$ if and only if $|z|$=$0$ if and only if $z$=$0$ and since the curve {$z$/ |$z$|=$2$} is a simple connected graph then $\int_{|z|=2} \frac{e^{senz}}{|z|^3}\,dz$ = 0

For (..) I know that the domain of analyticity of $\sqrt{z^2-1}$ is the whole complex plane minus $(-\infty,-1)$ $\cup$ $(1,\infty)$ over the branch $(0,2\pi)$ and since the curve ${|z|=\frac{1}{2}}$ is a simple closed curve that does not even touch the points of not analyticity of $\sqrt{z^2-1}$ then we have by cauchy theorem that $\int_{|z|=\frac{1}{2}} \sqrt{z^2-1}\,dz$= $0$

Are these arguments right? How can I filling up the gaps in the argumentation to formally write a proof?

Thanks for your help

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1 Answer 1

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The function $e^{senz}/|z|^3$ is not analytical. This can be proven by the identity theorem: The function $e^{senz}/z^3$ is holomorphic on $\mathbb C \setminus 0$ (quotient rule) and identical to $e^{senx}/x^3$ for $x \in (1, \infty)$. If the function $e^{senz}/|z|^3$ was holomorphic, we would have $e^{senz}/|z|^3 = e^{senz}/z^3$ for all $z \in \mathbb C$ by the identity theorem, since $e^{senx}/|x|^3 = e^{senx}/x^3$ on $(1, \infty)$ as well.

Therefore, it is advisable to observe that $$ \int_{|z| = 2} e^{senz}/|z|^3 dz = \frac{1}{8} \int_{|z| = 2} e^{senz} dz. $$ The latter integral is zero since $e^{senz}$ is holomorphic in the whole plane.

For the second integral $$ \int_{|z| = 1/2} \sqrt{z^2-1} dz = 0, $$ since your square root is analytic in the star domain $\mathbb C \setminus \left\{x + iy \middle| y = 0 \wedge x \ge 0\right\}$.

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