Computing Values of Regular Branches of Multivalued Functions. Laurent Series for Regular Branches.

Let $G$ be a simply connected domain not containing the points $z = 0$ and $z = \infty$, but containing the point $z = 1$. Determine how many distinct regular branches $\varphi (z)$ in the domain $G$, satisfying the indicated condition, the following multivalued functions admit:

Let $\varphi (z)$ be a regular branch of the multivalued function $\operatorname {Ln}(z+i)$ in a domain $G$, satisfying the condition $\varphi (1-i) = 0$. Find the value $\varphi (-1-i)$ in the cases when the domain $G$ is:

Let $G$ be the entire complex plane with cuts along the rays $[-\infty ,-1]$ and $[1,+\infty ]$, and let $\varphi (z)$ be a regular branch of the multivalued function $\operatorname {Ln}\left(1-z^{2}\right)$ in the domain $G$, satisfying the condition $\varphi (0) = 0$. Find:

Let $G$ be the entire extended complex plane with a cut along the segment $[-1,1]$. Denote by $\varphi_{1}(z)$ a regular branch of the multivalued function $\left\{ \sqrt{\frac{1+z}{1-z}}\right\}$ in the domain $G$, satisfying the condition $\varphi_{1}(+i 0) = 1$, and by $\varphi_{2}(z)$ a regular branch of the multivalued function $\operatorname {Ln} \frac{1+z}{1-z}$ in the domain $G$, satisfying the condition $\varphi_{2}(-i 0) = 0$. Find the quantities:

Let $G$ be the entire extended complex plane with a cut along the straight-line segment $[-i, i]$, and let $\varphi (z)$ be a regular branch of the multivalued function $\operatorname {Ln} \frac{1+i z}{1-i z}$ in the domain $G$, satisfying the condition $\varphi (1) = \pi i / 2$. Find the values:

Let $G$ be the entire complex plane with cuts along the segments $[-2,-1]$ and $[1,2]$, and let $\varphi (z)$ be a regular branch of the multivalued function $\left\{ \sqrt{\left(z^{2}-1\right)\left(z^{2}-4\right)}\right\}$ in the domain $G$, positive on the interval $(-1,1)$. Find the values:

Let $G$ be the entire complex plane with cuts along the segments $[-1, i]$ and $[-i, 1]$, and let $\varphi (z)$ be a regular branch of the multivalued function $\left\{ \sqrt{1-z^{4}}\right\}$, positive on the interval $(-1,1)$. Find the values:

Let $a$ and $b$ be two distinct finite points of the complex plane, and let $\gamma$ be some simple continuous curve going from the point $a$ to the point $b$. Denote by $G$ the entire complex plane with a cut along the curve $\gamma$, and by $\varphi (z)$ an arbitrary regular branch of the multivalued function $\operatorname {Ln} \frac{z-a}{b-z}$ in the domain $G$. Prove that for every point $z_{0}$ of the curve $\gamma$, distinct from the points $a$ and $b$, the equality $\varphi \left(z_{0}^{+}\right)-\varphi \left(z_{0}^{-}\right) = 2 \pi i$ holds, where the symbol $\varphi \left(z_{0}^{ \pm }\right)$ denotes the limit of the function $\varphi (z)$ as the point $z$ tends to the point $z_{0}$ from the right (from the left for the sign "$-$") of the curve $\gamma$.

Let $\varphi (z)$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{1-z^{2}}\right\}$ in a domain $G$, satisfying the condition $\varphi (0) = 1$. Find the value $\varphi (-3)$ in the cases when the domain $G$ is:

Let $P(z)$ be a polynomial. Prove that any function $f(z)$, regular in the entire complex plane with a cut along the positive part of the real axis, continuous up to the boundary of this domain, and satisfying, for all real $x > 0$, the condition

has the form

where $g(z)$ is a function regular in the entire complex plane, and $h(z)$ is a regular branch of the multivalued function $\operatorname {Ln} z$ in the plane with a cut along the positive part of the real axis.

Find the general form of a function $f(z)$, regular in the entire complex plane with a cut along the positive part of the real axis, continuous up to the boundary of this domain except at the point $z = 0$, and satisfying one of the conditions $(x > 0)$:

Let $G$ be the entire complex plane with a cut along the positive part of the real axis, and let $\varphi (z)$ be a regular branch of the multivalued function $\operatorname {Ln}(1+\sqrt{z})$, satisfying the condition $\varphi (-1) = \frac{1}{2} \ln 2+\frac{\pi i}{4}$. Find the value $\varphi (4-i 0)$.

Let $G$ be the entire complex plane with cuts along the rays $[-i \infty ,-i]$ and $[i,+i \infty ]$, and let the regular function $\varphi (z)$ be defined by the conditions

Find the values:

Let $\varphi (z)$ be a regular branch of the multivalued function $\operatorname {Ln}(\sqrt{z}+2 \sqrt{1-z})$, satisfying the condition

Find the value $\varphi (4)$ in the cases when the domain $G$ is:

Let $G$ be the entire complex plane with cuts along the rays $[-i \infty , 0]$ and $[1,1+i \infty ]$, and let the regular function $\varphi (z)$ be defined by the conditions

Find the value $\varphi \left(-e^{\pi }\right)$.

Let $G$ be the entire complex plane with a cut shown in Fig. 18.1, and let the regular function $\varphi (z)$ be defined by the conditions

Fig. 18.1

Find the values:

Let $G$ be the entire complex plane with a cut shown in Fig. 18.2, and let the regular function $\varphi (z)$ be defined by the conditions

Fig. 18.2

Find the values:

Let $G$ be the entire complex plane with a cut shown in Fig. 18.3, and let the regular function $\varphi (z)$ be defined by the conditions

Fig. 18.3

Find the values:

Prove that the multivalued function $\left\{ \sqrt[3]{1-z^{2}}\right\}$ does not admit the extraction of a regular branch in the domain $G = \left\{ z: 1 < \left|z\right| < \infty \right\}$.

Determine whether the multivalued functions given below admit the extraction of regular branches in the domains $G$ indicated in parentheses:

Determine, for what relationship between the numbers $\alpha_{1}$ and $\alpha_{2}$, the multivalued function

admits the extraction of a regular branch in the domain $1 < \left|z\right| < \infty$.

Determine, for what relationship between the numbers $\alpha_{1}, \alpha_{2}$ and $\alpha_{3}$, the multivalued function

admits the extraction of a regular branch in the domain $1 < \left|z\right| < \infty$.

Prove that in the plane with a cut along the negative part of the real axis there exists a regular branch of the multivalued function $\operatorname {Ln}(1-\sqrt{z})$, satisfying the condition $\varphi (2 i) = -\pi i / 2$.

Let $\varphi (z)$ be a regular branch of the multivalued function $\operatorname {Ln} z$ in the domain shown in Fig. 18.4, satisfying the condition $\varphi (1) = 0$. Find $\varphi^{\prime }(2)$. Expand the function $\varphi (z)$ in a Taylor series in a neighborhood of the point $z = -3$ in powers of $z+3$.

Fig. 18.4

Let $g(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{z}\right\}$ in the domain shown in Fig. 18.4, such that $g(-1) = -1$. Find $g(-2), g^{\prime }(-3)$. Expand $g(z)$ in a Taylor series in a neighborhood of the point $z = 2$ in powers of $z-2$.

Let $\varphi_{1}(z), \varphi_{2}(z)$ be the functions from Problem 4. Find $\varphi_{1}^{\prime }(-2), \varphi_{2}^{\prime }(2)$. Expand the function $\varphi_{2}(z)$ in a Taylor series in a neighborhood of the point $z = 4$ in powers of $(z-4)$. Expand $\varphi_{1}(z)$ and $\varphi_{2}(z)$ in Laurent series in a neighborhood of the point $z = \infty$.

Let $g(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[4]{z}\right\}$ in the plane with a cut along the ray $[0,+\infty )$, such that $g(1+i 0) = 1$. Find $g(1-i 0)$, $g(16-i 0), g(-16), g^{\prime }(-16), g^{\prime \prime }(-16)$.

Relying on the Taylor series expansion of the regular branches of the multivalued function $\left\{ (1+z)^{a}\right\}$ (see Example 2 in the reference material), prove the formulas for all the resulting regular branches $g(z), w(z), h(z)$ in the disk $\left|z\right| < 1$:

For all regular branches $h(z), w(z)$ of the multivalued functions in the disk $\left|z\right| < 1$, prove the formulas:

Find the Taylor series expansion in powers of $z$ in a neighborhood of the point $z = 0$ of the following functions, containing regular branches $g(z), h(z)$ of multivalued functions:

Using the fact that the regular function $f(z) \in \left\{ \frac{\operatorname {Arcsin} z}{\sqrt{1-z^{2}}}\right\}$ satisfies the differential equation

prove the formula

For the regular branch

prove the formula

Using the binomial series, prove the validity of the Laurent series expansion formulas for regular branches $f(z)$ of multivalued functions:

Verify that the following multivalued functions admit the extraction of regular branches in the annulus $G$, and expand all the regular branches in a Laurent series in powers of $z-a$ in the annulus $G$ (the point $a$ and the annulus $G$ are indicated in parentheses):

Let $g(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{z+9}\right\}$ in the plane with a cut along the curve $z = 9 e^{i t}, \quad -\pi \leqslant t \leqslant \frac{\pi }{2}$ and the ray $z = 9 i+t i$, $t \geqslant 0$, such that $\arg g(10) = \frac{2 \pi }{3}$. Compute $g(-8), g(-1), g(-9+8 i), g^{\prime }(0)$. Expand $g(z)$ in a Taylor series in a neighborhood of the point $z = 3$ in powers of $z-3$ and draw the largest domain in which this series converges to the function $g(z)$.

Let $h(z)$ be a regular branch of the multivalued function $\operatorname {Ln}(2-z)$ in the plane with a cut along the curve $z = 2 e^{i t}, 0 \leqslant t \leqslant \frac{3}{2} \pi$, and the ray $z = -2 i+t, t \geqslant 0$, such that $\mathrm{Im}\left(h(-3)\right) = 0$. Compute $h(-2-0), \quad h(-2+0), \quad h(2+i)$, $h^{\prime }(0)$. Expand the function $h(z)$ in a Taylor series centered at the point $z = -1$. Find the radius of convergence of this series. Draw the largest domain in which the series converges to the function $h(z)$.

Let $g(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{z(2-z)^{2}}\right\}$ in the plane with a cut along the segment $[0,2]$, such that $g(1+i 0) = 1$. Find $g(1-i 0), \quad g(-3), \quad g^{\prime }(3)$. Expand $g(z)$ in a Laurent series in powers of $z$ in a neighborhood of the point $z = \infty$.

Let $h(z)$ 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 $h(1+i 0) = 0$. Find $h(1-i 0)$, $g(-2), g^{\prime }(-3)$. Expand $h(z)$ in a Taylor series centered at the point $z = 5$ and in a Laurent series in powers of $z$ in a neighborhood of the point $z = \infty$. Find the domains of convergence of these series.

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

such that $h(0) = i \frac{\pi }{2}$. Expand $h(z)$ in a Laurent series in powers of $z$ in a neighborhood of the point $z = \infty$.

Let $h(z)$ be a regular branch of the multivalued function

in the plane with the point $z = 2$ removed and a cut along the semicircle $\left\{ z:\left|z\right| = 1, \mathrm{Re}\left(z\right) \geqslant 0\right\}$, such that $h(0) = -i \pi +\frac{1}{2}$. Expand $h(z)$ in a Laurent series in powers of $z$ in the annulus $1 < \left|z\right| < 2$.

Let $g(z)$ be a regular branch of the multivalued function

in the plane with the point $z = 3 i$ removed and a cut along the segment $[0,1]$, such that $g(-1) = \sqrt[3]{2}-\frac{1}{1+3 i}$. Expand $g(z)$ in a Laurent series in powers of $z$ in the annulus $1 < \left|z\right| < 3$.

Let $h(z)$ be a regular branch of the multivalued function $\operatorname {Ln}\left(z^{2}-z-6\right)$ in the plane with cuts along the lines $\left\{ \mathrm{Re}\left(z\right) = -2, \mathrm{Im}\left(z\right) \leqslant 0\right\}$ and $\left\{ \mathrm{Re}\left(z\right) = 3, \mathrm{Im}\left(z\right) \leqslant 0\right\}$, such that $h(4) = \ln 6$. Expand $h(z)$ in a Taylor series centered at the point $z = 1$ and find the radius of convergence of this series.

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln}\left(z^{2}-2 z\right)$ in the plane with cuts along the positive part of the imaginary axis and along the segment $[0,2]$ of the real axis, singled out by the condition

Compute $f(1+2 i)$. Expand $f(z)$ in a Taylor series centered at the point $z = 1+2 i$. Find the radius of convergence. Compute the sum of the series at $z = -1+\sqrt{3} i$.

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[3]{z^{4}+64}\right\}$ in the plane with cuts along the rays

singled out by the condition $f(2 \sqrt{2}) = 4 \sqrt[3]{2}$. Compute $f(0)$. Expand $f(z)$ in a Taylor series centered at the point $z = 0$. Find the radius of convergence. Compute the ratio of the sum of the series to $f(z)$ at $z = -\frac{5}{2}$.

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln}\left(z+i z^{2}\right)$ in the plane with cuts along the positive part of the real axis and along the segment $[0, i]$ of the imaginary axis, singled out by the condition

Compute $f\left(1+\frac{i}{2}\right)$. Expand $f(z)$ in a Taylor series centered at the point $z = 1+\frac{i}{2}$. Find the radius of convergence. Compute the sum of the series at $z = e^{-i \pi / 6}$.

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt[6]{z^{4}+4}\right\}$ in the plane with cuts along the rays $\left\{ z: z = i+t ; t \geqslant -1\right\}$ and $\left\{ z: z = -i+t ; t \leqslant 1\right\}$, singled out by the condition $f(\sqrt{2} i) = -\sqrt{2}$. Compute $f(0)$. Expand $f(z)$ in a Taylor series centered at the point $z = 0$. Find the radius of convergence. Compute the ratio of the sum of the series to $f(z)$ at $z = -\frac{6}{5} i$.

Let $f(z)$ be a regular branch of the multivalued function $\operatorname {Ln}\left(1+z^{2}\right)$ in the plane with a cut along the ray of the imaginary axis $[-i,+i \infty )$, with $\mathrm{Im}\left(f\right)\left(-\frac{1}{5}\right) = 0$. Expand $f(z)$ in a Taylor series centered at the point $z = 1$ and find the radius of convergence. Compute the sum of the series at $z = -\frac{1}{5}$.

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt{9-z^{2}}\right\}$ in the plane with a cut along the circular arc $\left\{ z:\left|z-4 i\right| = 5, \mathrm{Im}\left(z\right) \geqslant 0\right\}$, with $f(4 i) = 5$. Expand $f(z)$ in a Laurent series in powers of $z$ in a neighborhood of $z = \infty$ and find the domain of convergence. Compute the sum of the series at $z = 4 i$.