Conformal Mappings

What curve does the point $z$ of the complex plane describe, if $z = e^{(a+bi)t}$, where $z$ is expressed in terms of the real variable $t$, ranging from $-\infty$ to $\infty$, and the coefficients $a$, $b$ are real?

What curve does the point $z$ of the complex plane describe, if $z = at+be^{i\omega t}$, where $z$ is expressed in terms of the real variable $t$, ranging from $-\infty$ to $\infty$, and the coefficients $a$, $b$, $\omega$ are real?

What curve does the point $z$ of the complex plane describe, if $z = ae^{it}+a^{-1}e^{-it}$, where $z$ is expressed in terms of the real variable $t$, ranging from $-\infty$ to $\infty$, and the coefficient $a$ is real?

What curve does the point $z$ of the complex plane describe, if $z = a(1+e^{it})^{-2}$, where $z$ is expressed in terms of the real variable $t$, ranging from $-\infty$ to $\infty$, and the coefficient $a$ is real?

What curve does the point $z$ of the complex plane describe, if $z = a(1+it)e^{-it}$, where $z$ is expressed in terms of the real variable $t$, ranging from $-\infty$ to $\infty$, and the coefficient $a$ is real?

What line or domain in the plane of the variable $z$ is defined by the condition $\left|z-2\right| < \left|z\right|$?

What line or domain in the plane of the variable $z$ is defined by the condition $\left|z-1\right| \geqslant 2\left|z-i\right|$?

What line or domain in the plane of the variable $z$ is defined by the condition $\left|z^{2}-1\right| < 1$?

What line or domain in the plane of the variable $z$ is defined by the condition $\mathrm{Im}\left(z^{-1}\right) = 2$?

What line or domain in the plane of the variable $z$ is defined by the condition $0 < \arg \dfrac {z-i}{z+i} < \dfrac {\pi }{4}$?

What line or domain in the plane of the variable $z$ is defined by the condition $\left|2z\right| < \left|1+z^{2}\right|$?

Show that all the roots of the equation

lie on a single circle.

Show that the function $w = z^{2}$ maps the given domain of the plane $z = x+iy$ conformally and bijectively onto the indicated domain of the plane $w = u+iv$: the first quadrant of the plane $z$, onto the upper half-plane $w$.

Show that the function $w = e^{z}$ maps the given domain of the plane $z = x+iy$ conformally and bijectively onto the indicated domain of the plane $w = u+iv$: the strip $0 < y < \dfrac {\pi }{2}$, onto the first quadrant of the plane $w$.

Show that the function $w = \ln z$ maps the given domain of the plane $z = x+iy$ conformally and bijectively onto the indicated domain of the plane $w = u+iv$: the domain $x^{2}+y^{2} = 1$; $y > 0$, onto the strip $u > 0$, $0 < v < \pi$.

Show that the function $w = \cos z$ maps the given domain of the plane $z = x+iy$ conformally and bijectively onto the indicated domain of the plane $w = u+iv$: the strip $-\dfrac {\pi }{2} < x < \dfrac {\pi }{2}$, onto the half-plane $v > 0$.

Show that the function $w = \dfrac {1}{z+i}$ maps the given domain of the plane $z = x+iy$ conformally and bijectively onto the indicated domain of the plane $w = u+iv$: the half-plane $y \geqslant 0$, onto the disk $u^{2}+v^{2}+v \leqslant 0$.

Show that the function $w = \dfrac {i-z}{i+z}$ maps the given domain of the plane $z = x+iy$ conformally and bijectively onto the indicated domain of the plane $w = u+iv$: the first quadrant of $z$, onto the upper half-disk $\left|w\right| < 1$.

Into what do the lines $\left|z\right| = C$ and $\arg z = C$, and the region $\left|z\right| > C > 1$; $a < \arg z < \pi -a$, where $0 < a < \dfrac {\pi }{2}$, go under the conformal transformation $2w = z+z^{-1}$?

Into what is the upper half-plane $z$ transformed on the plane $w$, if $\cos w = \ch a\cos z$ and at $z = \dfrac {\pi }{2}$ we have $w = \dfrac {\pi }{2}$?

The function $w = \arctg z$ transforms the disk $\left|z\right| < 1$ into a strip in the plane $w$. Find the width of this strip.

Onto what domain is the disk $\left|z\right| < 1$ mapped by the function $w = a(nz-z^{n})$, where $a > 0$, $n > 1$ an integer? Find the area of this domain.

The function $w = \left(1+\dfrac {z}{n}\right)^{n}$ maps some domain $S_{n}$ onto the upper half-plane. Find $S_{n}$ and $\lim_{n\to \infty } S_{n}$.

The point $w = \dfrac {a^{2}}{z}$ is called the reflection of the point $z$ with respect to the circle $\left|z\right| = a$. Find the reflection, in this circle, of the half-plane $\mathrm{Re}\left(z\right) > 2a$.

Prove that under a linear fractional transformation $w = \dfrac {az+b}{cz+d}$, where $ad-bc \neq 0$, a point and its reflection with respect to a circle go into a point and its reflection with respect to the transformed circle.

Prove that the identity

gives the linear fractional function $w(z)$ that carries the points $z_{1}$, $z_{2}$, $z_{3}$ into the points $w_{1}$, $w_{2}$, $w_{3}$.

Prove that the disk $\left|z\right| < 1$ goes into itself under a linear fractional map only for $w = e^{i\theta }\cdot \dfrac {z-a}{1-\overline{a}z}$, where $\left|a\right| < 1$.

Find the conditions under which the linear fractional function $w = \dfrac {az+b}{cz+d}$ carries the disk $\left|z\right| < 1$ into the upper half-plane.

Find the function giving the conformal mapping of the disk $\left|z\right| < 1$ onto the half-plane $\mathrm{Im}\left(w\right) < 0$ such that the points $1$, $i$, $-i$ go into the points $w = +1, 0, -1$.

Find the function giving the conformal mapping of the same disk (problem 5617) into itself such that the points $z = -1, i, 1$ go into the points $w = -1, \dfrac {3i+4}{5}, 1$.

Find the function giving the conformal mapping of the digon bounded by arcs of circles meeting at the points $z = a$, $z = b$ at an angle $\dfrac {\pi }{\alpha }$, onto the half-plane $\mathrm{Im}\left(w\right) > 0$ such that the points $z = a$, $z = b$ go into $w = 0$, $w = \infty$.

Find the function giving the conformal mapping of the upper half-disk $\left|z\right| < 1$ onto the upper half-plane $\mathrm{Im}\left(w\right) > 0$ such that at $z = 0, 1, -1$ we have $w = 0, 1, -1$.

Find the function giving the conformal mapping of the quadrant $0 < \arg z < \dfrac {\pi }{2}$ onto the disk $\left|w\right| < 1$ such that at $z = 1+i, 0$ we have $w = 0, 1$.

Find the function giving the conformal mapping of the disk $\left|z\right| < 1$ onto the plane with a cut along the positive part of the real axis.

Find the function giving the conformal mapping of the plane with a cut along the whole real axis except the segment between the points $a$ and $b$, onto the region $\mathrm{Im}\left(w\right) > 0$.

Find the function giving the conformal mapping of the plane with a cut along the segment $a < z < b$ onto $\mathrm{Im}\left(w\right) < 0$.

Find the function giving the conformal mapping of the disk $\left|z\right| < 1$ with a slit along the segment $0 < z < 1$ onto the half-plane $\mathrm{Im}\left(w\right) > 0$.

Find the function giving the conformal mapping of the strip $0 < \mathrm{Re}\left(z\right) < a$ onto the half-plane $\mathrm{Im}\left(w\right) > 0$.

Find the function giving the conformal mapping of the domain $\left|z-a\right| > a$; $\left|z-b\right| < b$, where $0 < a < b$, onto the strip $0 < \mathrm{Im}\left(w\right) < \pi$.

Find the function giving the conformal mapping of the domain $0 < a < \left|z\right| < b$, $0 < \arg z < \pi$ onto a rectangle.

Find the function giving the conformal mapping of the domain $\left|z-3\right| > 9$; $\left|z-8\right| < 16$ onto the annulus between the circles: $\rho < \left|w\right| < 1$.

Find the function giving the conformal mapping of the interior of the ellipse $4x^{2}+5y^{2} = 20$ with a cut between the foci onto the interior of the annulus $1 < \left|w\right| < A$, and find $A$.

Find the function giving the conformal mapping of the interior of the parabola $y^{2} = 4x$ onto the half-plane $\mathrm{Im}\left(w\right) > 0$.

Find the function giving the conformal mapping of the interior of the hyperbola $x^{2}\sin^{2}\alpha -y^{2}\cos^{2}\alpha = \sin^{2} 2\alpha$ onto the half-plane $\mathrm{Im}\left(w\right) > 0$.

Find the function giving the conformal mapping of the interior of the right half of the lemniscate $z^{2} = 2a^{2}\cos 2\varphi$ onto the disk $\left|w\right| \leqslant 1$.

Prove that the function $w = \int_{0}^{z} z^{\frac{1}{3}-1}(1-z)^{\frac{1}{3}-1}\, dz$ maps the upper half-plane $z$ onto an equilateral triangle with side $\dfrac {\sqrt{3}}{2\pi }\Gamma^{3}\left(\dfrac {1}{3}\right)$.

Prove that the function $w = \int_{0}^{z} \dfrac {dz}{\sqrt{1-z^{4}}}$ maps the disk $\left|z\right| < 1$ onto a square with side

Prove that the function

maps the interior of the disk $\left|z\right| < 1$ onto a rhombus with angle $60°$ and side $\dfrac {1}{4\sqrt{\pi }}\Gamma \left(\dfrac {1}{3}\right)\Gamma \left(\dfrac {1}{6}\right)$.

Prove that the function

maps the interior of the disk $\left|z\right| < 1$ onto the area of a regular $n$-gon, whose side length equals

Onto what domain does the interior of the disk $\left|z\right| < 1$ get mapped by the function $\left(-1 < \lambda < 1-\dfrac {2}{n}\right)$

Find the function giving the conformal mapping of the disk $\left|z\right| < 1$ onto a regular five-pointed star.

Find the function giving the conformal mapping of the same disk (problem 5639) onto the exterior of a regular $n$-gon.

Find the function giving the conformal mapping of the half-plane $\mathrm{Im}\left(z\right) > 0$ onto the same half-plane with a cut between the points $w = 0$ and $w = i$.

Find the function giving the conformal mapping of the half-plane $\mathrm{Im}\left(z\right) > 0$ onto the plane with vertical cuts from the points $w = n\pi$ downward to infinity.

Find the function giving the conformal mapping of the disk $\left|z\right| < 1$ onto the disk $\left|w\right| < 1$ with a radial cut between the points $w = \dfrac {1}{2}$ and $w = 1$.

Find the function giving the conformal mapping of the plane with a cut along the axis $OX$ from the point $z = 1$ to $\infty$ onto the disk $\left|w\right| < 1$.