Residue at infinity

Compute:

Find $\mathop{\mathrm{res}}_{z = a} f(z)$, if:

Find the residues of the following functions at infinity:

Find the residues of the following functions at all their singular points and at infinity:

Let the function $g(z)$ be regular at the point $z = \infty$, and let the function $h(z)$ have a zero of multiplicity 1 at the point $z = \infty$. Prove that

Prove that for every even function $f(z)$

(assuming that the residues written make sense).

Let the function $f(z)$ be regular for $R < \left|z\right| < \infty$, with $\mathop{\mathrm{res}}_{z = \infty } f(z) = 0$. Prove that the function $f(z)$ has an antiderivative, regular for $R < \left|z\right| < \infty$.

Compute the integrals:

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln} \frac{z+1}{3-z}$ in the plane with a cut along the segment $[-1,3]$, such that $f(1+0 i) = 0$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[4]{z^{3}(6-z)}\right\}$ in the plane with a cut along the segment $[0,6]$, such that $f(3+0 i) = 3$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln} \frac{5-z}{3+z}$ in the plane with a cut along the segment $[-3,5]$, such that $f(1+0 i) = 0$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{z^{2}(4-z)}\right\}$ in the plane with a cut along the segment $[0,4]$, such that $f(2+0 i) = 2$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln} \frac{3+i z}{z-1}$ in the plane with a cut along the segment $[1,3 i]$, such that $f(0) = \ln 3-\pi i$. Compute $f(\infty )$ and the integral

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[4]{z^{3}(2 i-z)}\right\}$ in the plane with a cut along the segment $[0,2 i]$, such that

Compute $f(i+0)$ and the integral

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln} \frac{z+i}{2-z}$ in the plane with a cut along the segment $[-i, 2]$, such that $f(0) = \ln \frac{1}{2}+\frac{5}{2} \pi i$. Compute $f(\infty )$ and the integral

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{z^{2}(i-z)}\right\}$ in the plane cut along the segment $[0, i]$ such that $f(1) = \sqrt[6]{2} e^{i \pi / 4}$. Compute $f\left(\frac{i}{2}+0\right)$ and the integral

Let $g(z)$ be a regular branch of the multivalued function $\operatorname {Ln} \frac{2 i-z}{z+1}$ in the plane cut along the curve $\gamma = \gamma_{1} \cup \gamma_{2}$, where

such that $g(0) = \ln 2-\frac{3 \pi i}{2}$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{z^{2}(i-z)}\right\}$ in the plane cut along the curve $\gamma = \gamma_{1} \cup \gamma_{2}$, where

such that $f(-i) = \sqrt[3]{2} e^{i 7 \pi / 6}$. Compute the integral

Let $g(z)$ be a regular branch of the multivalued function $\operatorname {Ln} \frac{1-z}{i z+1}$ in the plane cut along the curve

such that $g(0) = -4 \pi i$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[4]{z^{2}(2 i+z)^{2}}\right\}$ in the plane cut along the curve $\gamma = \gamma_{1} \cup \gamma_{2}$, where

such that $f(-3 i) = \sqrt{3} e^{\pi i}$. Compute the integral

Let $h(z)$ be a regular branch of the multivalued function $\operatorname {Ln} \frac{2+i z}{2+z}$ in the plane cut along the curve

such that $h(\infty ) = \frac{1}{2} \pi i$. Also compute the integral

Let $g(z)$ be a regular branch of the multivalued function $\left\{ \sqrt{2 z^{2}+1}\right\}$ in the plane cut along the curve $\gamma = \left\{ z:\left|z\right| = \frac{1}{\sqrt{2}}, \mathrm{Re}\left(z\right) \geqslant 0\right\}$, where $g(0) = 1$. Compute the integral

Let $g(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{(z-2)^{2}(2 i-z)}\right\}$ in the plane cut along the segment $[2 i, 2]$ such that the principal part of its Laurent series at $\infty$ equals $e^{-i \pi / 3} z$. Compute the integral

Let $h(z)$ be a regular branch of the multivalued function $\operatorname {Ln} \frac{3+z}{i z-3}$ in the plane cut along the curve

such that $h(\infty ) = -\frac{5}{2} \pi i$. Compute the integral

Let $h(z)$ be a regular branch of the multivalued function $\operatorname {Ln} \frac{2+z}{i z-1}$ in the plane cut along the curve

such that $\mathrm{Im}\left(h(\infty )\right) = \frac{3 \pi }{2}$. Find $h(0)$ and compute the integral

Let $g(z)$ be a regular branch of the multivalued function $\left\{ \sqrt{1+2 z^{2}}\right\}$ in the plane cut along the curve

such that $g(0) = 1$. Let $f(z) = \frac{z}{(g(z)+3)^{2}}$. Find $\mathop{\mathrm{res}}_{z = \infty } f$ and compute the integral

Let $g(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{z^{2}+1}\right\}$ in the complex plane cut along the curve $\gamma = \gamma_{1} \cup \gamma_{2}$, where

with $g(0) = e^{-2 \pi i / 3}$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln}\left(z^{2}-1\right)$ in the complex plane cut along the curve $\gamma = \gamma_{1} \cup \gamma_{2}$, where

with $\mathrm{Im}\left(h(-2 i)\right) = 3 \pi$. Compute the integral

Let $g(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[4]{z^{2}-1}\right\}$ in the complex plane cut along the curve $\gamma = \gamma_{1} \cup \gamma_{2}$, where

with $g(2) = \sqrt[4]{3}$. Compute the integral

Let $h(z)$ be a regular branch of the multivalued function $\operatorname {Ln}\left(z^{2}+1\right)$ in the complex plane cut along the curve $\gamma = \gamma_{1} \cup \gamma_{2}$, where

with $\mathrm{Im}\left(h(0)\right) = 0$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln}\left(\frac{z-1}{z+1}\right)$ in the domain $G = \left\{ z:\left|z\right| > 1\right\}$ such that $f(\infty ) = 0$. Prove that the multivalued function $\left\{ \sqrt{f(z)+4 i}\right\}$ splits in $G$ into regular branches.

Let $g(z)$ be a regular branch of the function $\left\{ \sqrt{f(z)+4 i}\right\}$ in $G$ such that $g(\infty ) = -\sqrt{2}(i+1)$. Compute the integral $\oint_{\left|z\right| = 2} \frac{d z}{g(z)}$.

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt{\frac{3-z}{3+z}}\right\}$ in the domain $G = \left\{ z:\left|z\right| > 3\right\}$ such that $f(\infty ) = -i$. Prove that the multivalued function $\operatorname {Ln}(f(z)-i)$ splits in $G$ into regular branches.

Let $g(z)$ be a regular branch of the function $\operatorname {Ln}(f(z)-i)$ in $G$ such that $g(\infty ) = \ln 2-i \frac{\pi }{2}$. Compute the integral $\oint_{\left|z\right| = 4} \frac{d z}{g(z)}$.

Compute the integrals: