Taylor and Laurent expansions of regular branches of multivalued functions

Find a Laurent series that converges in the annulus $1<\left|z\right|<2$ to a branch of the function $\log \left(\dfrac {z(2-z)}{1-z}\right)$.

Let $f(z)$ be the ``branch'' of $\log z$ defined off the negative real axis so that $f(1) = 0$.

Expand in a series of powers of $z$ the branch of the function $\sqrt{z+i}$ whose value at $z = 0$ is $\dfrac {1+i}{\sqrt{2}}$.

Expand the function $\dfrac {1}{z-2}\ln \dfrac {z-i}{z+i}$ in a Laurent series for $1 < \left|z\right| < 2$, taking the value of the logarithm at $z = 2i$ to be real.

Expand the function $\dfrac {1}{z-2}\ln \dfrac {z-i}{z+i}$ in a Laurent series for $\left|z\right| > 2$ (the same function as in the previous problem, for a different domain condition, taking the value of the logarithm at $z = 2i$ to be real).

There is a region $\Omega$ that $\exp (\Omega ) = D(1 ; 1)$. Show that $\exp$ is one-to-one in $\Omega$, but that there are many such $\Omega$. Fix one, and define $\log z$, for $\left|z-1\right| < 1$, to be that $\dot{w} \in \Omega$ for which $e^{w} = z$. Prove that $\log^{\prime }(z) = 1 / z$. Find the coefficients $a_{n}$ in

and hence find the coefficients $c_{n}$ in the expansion

In what other discs can this be done?

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 $z = \infty$.

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 $z = 4 i$.

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 real axis $(-\infty , 1]$, with $\mathrm{Im}\left(f\right)\left(\frac{i}{5}\right) = 0$.

Expand $f(z)$ in a Taylor series centered at the point $z = -i$. Find the radius of convergence. Compute the sum of the series at $z = \frac{i}{5}$.

Let $f(z)$ be a regular branch of the multivalued function $\left\{ \sqrt{z^{2}+16}\right\}$ in the plane with a cut along the circular arc $\left\{ z:\left|z+3\right| = 5, \mathrm{Re}\left(z\right) \geqslant 0\right\}$, with the principal part of the Laurent series of the function $f(z)$ in a neighborhood of $z = \infty$ equal to $z$. Expand $f(z)$ in a Taylor series centered at the point $z = 0$, find the radius of convergence. Compute the sum of the series at $z = 3$.

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