Integrals of regular branches

Let $C$ denote the positively oriented circle $\left|z\right| = 2$, $z \in \mathbb {C}$. Evaluate the integral

where the branch of the square root is chosen so that $\sqrt{4} - 1 > 0$.

Let $f(z)$ be holomorphic in the unit disk $\left|z\right| < 1$. Prove that

where respective integration goes along the straight line from $0$ to $1$ and along the positively oriented unit circle starting from the point $z=1$. The branch of $\log$ is chosen to be real for positive $z$.

We suppose that the function $f(z)$ is regular in the disk $\left|z\right|\leq 1$. Then

We integrate with respect to $x$ along a straight line from $0$ to $1$, with respect to $z$ along the positively oriented unit circle. We start at the point $z=1$ with the branch of $\log z$ that is real, and the branch of $z^{k}$ that is positive, for positive $z$.

The function $f(z)$ is regular in the unit disk $\left|z\right|\leq 1$ and satisfies the condition

Let $k>-1$. Then

For all branches $h_{k}(z)$ or $g_{k}(z)$, regular in the domain $G$, of the indicated multivalued functions, compute the integrals over the positively oriented boundary $\partial G$ of the domain $G$:

Verify that the multivalued functions admit, in the domain $G$, the extraction of a regular branch satisfying the given conditions, and compute the integral of this branch over the positively oriented boundary $\partial G$ of the domain $G$:

Compute the integral of branches, regular in the domain $G$, of multivalued functions over the boundary $\partial G$ of the domain $G$:

where $G = \left\{ z \in \mathbb {C}: z \notin (-\infty , 0]\right\} , \quad g(z) \in \left\{ \sqrt{z}\right\} , \quad g(1) = 1, \quad h(z) \in \operatorname {Ln} z$, $h(1) = 0$.

Compute the integrals of branches, regular in the domain $G$, of multivalued functions over the boundary $\partial G$ of the domain $G$:

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{z+9}\right\}$ in the complex plane with a cut along the curve $\left\{ z: z = 9 e^{i t},-\pi \leqslant t \leqslant \frac{\pi }{2}\right\}$, and the ray $\left\{ z: z = 9 i+t i, t \geqslant 0\right\}$, such that $\arg f(10) = \frac{2 \pi }{3}$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln}(z+3)$ in the complex plane with a cut along the curve $\left\{ z: z = 3 e^{i t},-\pi \leqslant t \leqslant \frac{\pi }{2}\right\}$ and the ray $\left\{ z: z = 3 i-t, t \geqslant 0\right\}$, such that $\mathrm{Im}\left(f(4)\right) = 2 \pi$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[4]{z+16}\right\}$ in the complex plane with a cut along the curve $\left\{ z: z = 16 e^{i t},-\frac{\pi }{2} \leqslant t \leqslant \pi \right\}$, and the ray $\left\{ z: z = -16 i-t i, t \geqslant 0\right\}$, such that $\arg f(20) = -\frac{\pi }{2}$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln}(z+5)$ in the complex plane with a cut along the curve $\left\{ z: z = 5 e^{i t},-\frac{\pi }{2} \leqslant t \leqslant \pi \right\}$ and the ray $\left\{ z: z = -5 i-t, t \geqslant 0\right\}$, such that $\mathrm{Im}\left(f(6)\right) = -2 \pi$. Compute the integral

Let $f(z), f(0) = \ln 2+i \pi ,$ be a regular branch of the multivalued function $\operatorname {Ln}(z-2)$ in the plane with a cut along the segment $[1,2]$ and the ray $\left\{ z: z = 1+i y,-\infty < y \leqslant 0\right\}$. Compute the integral

Let $f(z), \quad f(1-i) = i,$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{z-1}\right\}$ in the plane with a cut along the segment $[1,2]$ and the ray $\left\{ z: z = 2-i y, 0 \leqslant y < +\infty \right\}$. Compute the integral

Let $f(z), f(0) = \ln 4-i \pi ,$ be a regular branch of the multivalued function $\operatorname {Ln}(z-4)$ in the plane with a cut along the segment $[3,4]$ and the ray $\left\{ z: z = 3+i y, 0 \leqslant y < +\infty \right\}$. Compute the integral

Let $f(z), f(-2) = 1-i,$ be a regular branch of the multivalued function $\left\{ \sqrt[4]{z-2}\right\}$ in the plane with a cut along the segment $[1,2]$, and the ray $\left\{ z: z = 1+i y, 0 \leqslant y < +\infty \right\}$. Compute the integral

Let $f(z), f(2) = 0$, be a regular branch of the multivalued function $\operatorname {Ln}(z-1)$ in the plane with a cut along the segments $[0,1],[0, i]$, and the ray $\left\{ z: \mathrm{Im}\left(z\right) = 1, \mathrm{Re}\left(z\right) \geqslant 0\right\}$. Compute the integral

Let $f(z), \quad f(9) = 2$, be a regular branch of the multivalued function $\left\{ \sqrt[3]{z-1}\right\}$ in the plane with a cut along the segments $[0,1],[0,-i]$, and the ray $\left\{ z: \mathrm{Im}\left(z\right) = -1, \mathrm{Re}\left(z\right) \geqslant 0\right\}$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{2 z-8}\right\}$ in the complex plane with a cut along the ray $\left\{ z: z = 4-i t, t \in [0,+\infty )\right\}$, with $f(8) = -1-i \sqrt{3}$. Compute the integral

where $\partial D$ is the boundary of the disk $D = \left\{ z:\left|z-2\right| < \frac{3}{2}\right\}$.

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln} \frac{z^{2}+2 z}{4}$ in the plane with a cut along the ray $[-2,+\infty )$ of the real axis, with $\mathrm{Im}\left(f(-4)\right) = 0$. Compute the integral

where the domain $D$ consists of the points of the disk $\left|z+2\right| < 4$ whose distance from the cut is greater than 1.

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt{z^{2}-4}\right\}$ in the plane with a cut along the semicircle $\left\{ z:\left|z\right| = 2\text{, Re }z \geqslant 0\right\}$, with the principal part of the Laurent series of the function $f(z)$ in a neighborhood of $z = \infty$ equal to $z$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln}\left(z^{2}-z\right)$ in the plane with a cut along the positive real half-axis, with $\mathrm{Im}\left(f(-2)\right) = 0$. Compute the integral

where $D$ is the domain consisting of the points of the disk $\left|z\right| < 2$ whose distance from the cut is greater than $1 / 2$.

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 $\operatorname {Ln} \frac{1-z}{1+z}$ in the plane with a cut along the arc: $\left\{ z:\left|z\right| = 1, \mathrm{Im}\left(z\right) \leqslant 0\right\}$, such that $f\left(\frac{i}{\sqrt{3}}\right) = -\frac{\pi i}{3}$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt{2 z^{2}+1}\right\}$ in the plane with a cut along the arc: $\left\{ z:\left|z\right| = \frac{1}{\sqrt{2}}, \mathrm{Re}\left(z\right) \geqslant 0\right\}$, such that $f(0) = 1$.

Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln} \frac{2 i-z}{2 i+z}$ in the plane with a cut along the arc: $\left\{ z:\left|z\right| = 2, \mathrm{Re}\left(z\right) \leqslant 0\right\}$, such that $f(2) = \frac{\pi }{2} i$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt{z^{2}+3}\right\}$ in the plane with a cut along the arc: $\left\{ z:\left|z\right| = \sqrt{3}, \mathrm{Re}\left(z\right) \leqslant 0\right\}$, such that $f(1) = 2$. Compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[4]{z^{2}-1}\right\}$ in the plane with a cut along the curve

such that $f(0) = e^{\pi i / 4}$. Let $S(z) = \sum_{k = 0}^{+\infty } A_{k}(z-2 i)^{k}$ be the function, regular in its disk of convergence, coinciding with $f(z)$ in a neighborhood of the point $z = 2 i$. Find the radius of convergence of the series $S(z)$ and compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{z(z-1)}\right\}$ in the plane with cuts along the curves $\gamma_{1}$ and $\gamma_{2}$, where

such that $f\left(\frac{1}{2}\right) = -\sqrt[3]{\frac{1}{4}}$. Let $S(z) = \sum_{k = 0}^{+\infty } A_{k}(z+3 i)^{k}$ be the function, regular in its disk of convergence, coinciding with $f(z)$ in a neighborhood of the point $z = -3 i$. Find the radius of convergence of the series $S(z)$ and compute the integral

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[4]{1-z^{2}}\right\}$ in the plane with a cut along the curve

such that $f(0) = 1$. Let $S(z) = \sum_{k = 0}^{+\infty } A_{k}(z-3 i)^{k}$ be the function, regular in its disk of convergence, coinciding with $f(z)$ in a neighborhood of the point $z = 3 i$. Find the radius of convergence of the series $S(z)$ and compute the integral

In the complex plane with cuts along the curves $\gamma_{1}$ and $\gamma_{2}$, where

consider a regular branch $f(z)$ of the multivalued function $\left\{ \sqrt[3]{z(z+2)}\right\}$, such that

Let $S(z) = \sum_{k = 0}^{+\infty } A_{k}(z-3 i)^{k}$ be the function, regular in its disk of convergence, coinciding with $f(z)$ in a neighborhood of the point $z = 3 i$. Find the radius of convergence of the series $S(z)$ and 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