Singular Points of a Function. Taylor and Maclaurin Series

Find the singular points of the function $\dfrac {z}{z^{2}+1}$.

Find the singular points of the function $\dfrac {1}{z-z^{3}}$.

Find the singular points of the function $\dfrac {1}{\sin z}$.

Find the singular points of the function $\dfrac {1}{e^{z}-1}-\dfrac {1}{z}$.

Find the singular points of the function $\sin \dfrac {1}{z}$.

Find the singular points of the function $\sqrt{1-z^{2}}$.

Find the singular points of the function $\sqrt{1-z^{3}}$.

Construct the lines of equal modulus and equal argument for the function $w = e^{\frac{1}{z}}$.

Show that for any $A \neq 0$ the equation $e^{1/z} = A$ has infinitely many roots in any given neighborhood of the point $z = 0$.

Show that in the same region the equation $\cos \dfrac {1}{z} = A$ has infinitely many roots for any value of $A$.

Study the change in $\arg \sqrt[n]{z-a}$ as the point $z$ traces a simple closed contour going around the point $z = a$ in the positive direction.

Study the change in the function $\sqrt{1-z^{2}}$ as the point $z$ traces a simple closed contour of one of three kinds:

Find the value at the point $z = -e$ of the function $w = z^{z} = e^{z\ln z}$, if $\ln 1 = 0$ and if $w$ varies continuously as $z$ moves in the upper half-plane, or as $z$ moves in the lower half-plane.

Find the value at $z = e^{-\varphi i}$ of the branch of the function $\sqrt{z-e^{\varphi i}}$ which at $z = 0$ has the value $ie^{\frac{\varphi }{2}i}$ and which is continued continuously as $z$ moves along the radius from the origin to the point $z = e^{-\varphi i}$.

In the $z$-plane a cut is made along the lines: $y = 0$; $\left|x\right| > 1$ and $x^{2}+y^{2} = 1$; $y > 0$; the function $\ln (1-z^{2})$ is zero at $z = 0$. Find its value at $z = 3$.

Show that the function $w = \ln (z+\sqrt{z^{2}-1})$ is single-valued in the plane with cuts along the real axis for $-\infty < x < -1$ and $1 < x < \infty$. Find the values of $w$ at $z = +i$ and $z = -i$, given that on the upper edge of the right-hand cut $w$ is positive.

Using the series expansion for $\dfrac {1}{\cos z}$ and $z\ctg z$, prove the identity:

$E_{n}$ are the Euler numbers ($E_{0} = 1$).

Using the series expansion for $\dfrac {1}{\cos z}$ and $z\ctg z$, prove the identity:

$B_{n}$ are the Bernoulli numbers.

Using the series expansion for $\dfrac {1}{\cos z}$ and $z\ctg z$, prove the identity:

Using the series expansion for $\dfrac {1}{\cos z}$ and $z\ctg z$, prove the identity:

Using the series expansion for $\dfrac {1}{\cos z}$ and $z\ctg z$, prove the identity:

Find the first six coefficients in the expansion of the functions, and determine the domains of convergence of these series:

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

Find the circle of convergence for the expansion in powers of $z$ of the functions $\dfrac {\sin \pi z^{2}}{\sin \pi z}$ and $\dfrac {\cos 2\pi z^{2}}{\cos \pi z}$.

Expand the function $\dfrac {z}{(z^{2}-4)(z^{2}-1)}$ in a Laurent series for $1 < \left|z\right| < 2$.

Expand the function $\dfrac {1}{(z^{2}-1)(z^{2}-4)^{2}}$ in a Laurent series for $1 < \left|z\right| < 2$.

Expand the function $\dfrac {1}{(z^{2}-1)(z^{2}-4)^{2}}$ in a Laurent series for $\left|z\right| > 2$ (the same function as in the previous problem, for a different domain condition).

Expand the function $\ctg z$ in a Laurent series:

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).

Expand the function $z^{5}e^{1/z}$ in a Laurent series.

Expand the function $e^{z+1/z}$ in a Laurent series.

Prove the theorem: if the series $\sum_{n=0}^{\infty } a_{n}z^{n}$ has only one singular point on the boundary of its circle of convergence, namely a pole of order $m$, then the identity $a_{n} = \dfrac {An^{m-1}}{z_{0}^{n}}\left(1+\varepsilon (n)\right)$ holds, where $\varepsilon (n) \to 0$ as $n \to \infty$, $A$ is a constant, and $z_{0}$ is the pole.

Prove the theorem: if the radius of convergence of the power series $\sum a_{n}z^{n}$ equals one, and the only singular points on the circle of convergence are poles whose order does not exceed $m$, then the coefficients of the series do not exceed $An^{m-1}$, where $A$ is a constant.

Prove the theorem: if for the series $\sum a_{n}z^{n}$ all $a_{n} \geqslant 0$, and the radius of the disk equals $1$, then $z = 1$ is a singular point.

Prove the theorem: if the radius of convergence of the series $\sum a_{n}z^{n}$ and $\sum \mathrm{Re}\left(a_{n}\right)\cdot z^{n}$ equals $1$ and all $\mathrm{Re}\left(a_{n}\right) \geqslant 0$, then the point $z = 1$ is a singular point for the sums of these series.