Mappings and Vector Fields

Are the functions

analytic?

Assume that $f(z)$ is analytic, use the notation

as above and use subscripts to denote partial derivatives in the usual way. Verify that

Assume that $f(z)$ is analytic, use the notation

as above and use subscripts to denote partial derivatives in the usual way. Verify that

(continued). Prove that

(continued). Prove that

(continued). Let $\varphi (x,y)$ and $\psi (x,y)$ denote functions of the two real variables $x$ and $y$ having continuous derivatives of first order; they can also be considered as functions of $u$ and $v$ where $f'(z) \neq 0$. Verify that

(continued). Let $\varphi (x,y)$ and $\psi (x,y)$ denote functions of the two real variables $x$ and $y$ having continuous derivatives of first order; they can also be considered as functions of $u$ and $v$ where $f'(z) \neq 0$. Verify that

(continued). If $\varphi (x,y)$ has continuous derivatives of the second order, also

(continued). If $\varphi (x,y)$ has continuous derivatives of the second order, also

(continued). Assume that $a, b, c$ and $d$ are real constants,

and consider

Then

(continued). Assume that $a, b, c$ and $d$ are real constants,

and consider

Then

Find the analytic function of $z$ that vanishes for $z = 0$ and has the real part

We denote by $a$ and $b$, $a < b$, two fixed real numbers, by $z$ a variable point in the half-plane $\mathrm{Im}\left(z\right) > 0$ and by $\omega$ the variable angle under which the interval $[a,b]$ is seen from the point $z$. If possible find an analytic function the real part of which is $\omega$.

Show that for any analytic function $f(z) = f(x+iy)$

Show that for any analytic function of $z = x+iy$

We consider an orthogonal coordinate system $\xi , \eta , \zeta$ in three dimensional space. An arbitrary point $(\xi ,\eta ,\zeta )$ of the sphere $\xi^{2}+\eta^{2}+\zeta^{2}=1$ (unit sphere) is projected from the point $(0,0,1)$ (north-pole of the sphere) into the plane $\zeta = 0$ (equatorial plane). Let the projection be $(x,y,0)$. Express $x+iy$ in terms of $\xi ,\eta ,\zeta$ and $\xi ,\eta ,\zeta$ in terms of $x$ and $y$. (Stereographic projection.)

(continued). Let the point $P$ on the plane $\zeta =0$ be the stereographic projection of the point $P'$ on the unit sphere. A rotation through the angle $\pi$ of the unit sphere around the $\xi$-axis moves the point $P'$ to the point $P''$. This point $P''$ is then projected stereographically into the point $P'''$ of the $\xi ,\eta$-plane. Let the point $P$ have the coordinates $x,y,0$ and the point $P'''$ the coordinates $u,v,0$. Express $u+iv$ in terms of $x+iy$.

We introduce on the unit sphere the geographic coordinates $\theta$ and $\varphi$ (longitude and latitude) whereby

The sphere is described by

We consider now the circular cylinder tangent to the unit sphere $\xi^{2}+\eta^{2}+\zeta^{2}=1$ along the equator (great circle in the plane $\zeta =0$). Imagine a system of coordinates $(u,v)$ on the cylinder that becomes a cartesian system when the cylinder is unrolled. Let the point $u=0$, $v=0$ coincide with $(1,0,0)$, the positive $u$-axis be a generatrix pointing upwards and let the $v$-axis on the cylinder coincide with the equator. The values of $v$ vary on the cylinder in the same sense as $\theta$ from $-\pi$ to $\pi$. The points obtained in this way fill out an infinite strip of width $2\pi$ with the $u$-axis as center-line when the cylinder is unrolled.

Mercator's projection establishes a conformal one to one correspondence between the unit sphere and the $u,v$-cylinder (unrolled into a strip). The point on the cylinder corresponding to the point $\varphi ,\theta$ on the sphere has the coordinates

Into which lines does the Mercator projection transform the meridians and parallel circles? What are their images under the stereographic projection?

(continued). The point $P$ of the unit sphere is stereographically projected onto the point $(x,y,0)$ of the plane $z=0$ and the image of $P$ under Mercator's projection is $u,v$ on the cylinder. Express $x+iy$ in terms of $u+iv$.

Suppose $z=e^{w}$. To what curves in the $z$-plane do the two families of lines $\mathrm{Re}\left(w\right)=\text{const.}$ and $\mathrm{Im}\left(w\right)=\text{const.}$ in the $w$-plane (which are orthogonal to each other) correspond?

Along which curves of the $z$-plane is the real part of $z^{2}$ constant? Along which curves is the imaginary part constant? The two families of curves form an orthogonal system; why?

Which curves in the $z$-plane are transformed by $w=\sqrt{z}$ into the lines $\mathrm{Re}\left(w\right)=\text{const.}$ in the $w$-plane? Same question for $\mathrm{Im}\left(w\right)=\text{const.}$

The mapping $w=\cos z$ transforms the line $\mathrm{Re}\left(z\right)=\text{const.}$ of the $z$-plane into hyperbolas, the lines $\mathrm{Im}\left(z\right)=\text{const.}$ into ellipses of the $w$-plane.

Consider the function $z=w+e^{w}$, $z=x+iy$, $w=u+iv$. Find the equations of the curves in the $x,y$-plane that are mapped onto the lines $u=\text{const.}$ and $v=\text{const.}$ respectively. What corresponds to the lines $v=0$, $v=\pi$?

Given the function $w=e^{z}$ compute the area of the image of the square $a-\varepsilon \leq x \leq a+\varepsilon$, $-\varepsilon \leq y \leq \varepsilon$, $0<\varepsilon <\pi$, $z=x+iy$. Find the ratio of the two areas and its limit as $\varepsilon$ converges to $0$.

The function $w=\cos z$, $z=x+iy$, yields a one to one conformal mapping of the rectangle

onto a domain bounded by parts of confocal ellipses and hyperbolas [problem 67]. Compute the area of this domain.

Consider the function $w=z^{2}$. What is the locus of those points at which the linear enlargement equals some given constant? Analogous question for the rotation.

Let $a$ be an increasing positive parameter. Determine the region onto which the function $w=e^{z}$, $z=x+iy$, maps the variable square $-a<x<a$, $-a<y<a$. Up to what value of $a$ is this region covered only once? For what values of $a$ is the image covered exactly $n$ times?

We examine the image of the closed disk $\left|z\right|\leq r$ under the function $w=e^{z}$. Suppose $r$ is continuously increasing. There is on the ray $\arg w=\alpha$ a point that is covered by the image of the disk growing with $r$ at least as often as any other point of that ray for all values of $r$. Where is this point?

The function $w=z^{2}+2z+3$ is schlicht in the open disk $\left|z\right|<1$.

The function $w=z^{2}$ is schlicht in the upper half-plane $\mathrm{Im}\left(z\right)>0$ and maps it onto the $w$-plane cut along the non-negative real axis.

Let $\alpha$ be real and $\left|a\right|<1$. The function

maps the unit disk $\left|z\right|\leq 1$ univalently onto itself. [Problem 5.] What is the locus of those points at which the linear enlargement equals some given constant?

Assume that $C$ is a circle inside the unit disk. Then there exists a transformation of the unit disk onto itself of the type

that maps the circle $C$ onto a circle centred at the origin.

Find a function that transforms the upper half-plane $\mathrm{Im}\left(z\right)>0$ onto the disk $\left|w\right|<1$ so that $z=i$ is transformed into $w=0$.

The function $w=\frac{1}{2}\left(z+\frac{1}{z}\right)$ is schlicht in the open unit disk and maps it onto the $w$-plane cut open along the real line segment $-1\leq w\leq 1$. What curves correspond to the rays starting from the origin? What is the image of the unit circle?

Find a function that maps the annulus $0<r_{1}<\left|z\right|<r_{2}$ onto the area bounded by the two confocal ellipses

Transform the upper half of the unit disk $\left|z\right|<1$, $\mathrm{Im}\left(z\right)>0$ onto the upper half-plane [problem 79]. At which points of the $z$-plane is the linear enlargement equal to $\frac{1}{2}$? At which points is the rotation $\pm \frac{\pi }{2}$?

Map the upper half of the unit disk $\left|z\right|<1$, $\mathrm{Im}\left(z\right)>0$ onto the $w$-plane cut along the non-negative real axis in such a way that $z=0$ corresponds to $w=0$, $z=1$ to $w=1$, $z=i$ to $w+\infty$. Which point corresponds to $z=-1$?

Let $0\leq \alpha <\beta <2\pi$. The function

maps the sector $\alpha <\arg z<\beta$ onto the $w$-plane cut open along the non-negative real axis.

Let $0\leq \alpha <\beta <2\pi$. Map the circular sector

onto the unit disk $\left|w\right|<1$.

Show that

($\varphi (x,y)$ is the potential of the vector field; the curves $\varphi (x,y)=\text{const.}$ are the level lines.) Furthermore

($\psi (x,y)$ is the conjugate potential or stream function or stream potential; the lines $\psi (x,y)=\text{const.}$ are the stream lines or the lines of force, depending on the physical interpretation of the vector $\overline{w}$.)

The lines $\varphi (x,y)=\text{const.}$ and the lines $\psi (x,y)=\text{const.}$ are orthogonal to each other.

We have

Connect two points $z_{1}=x_{1}+iy_{1}$ and $z_{2}=x_{2}+iy_{2}$ in the vector field by a curve $L$ whose line element forms the angle $\tau$ with the positive $x$-axis. Prove that

i.e. the line integral of the tangential component of $\overline{w}$ is equal to the potential difference (work).

Using the same notation as in problem 88 establish the relation

i.e. the line integral of the normal component of $\overline{w}$ is equal to the change of the stream function (flux of force). (The normal to the curve $L$ is pointing to the right as one moves from $z_{1}$ to $z_{2}$:

The steady flow of an incompressible fluid of constant density $\varrho$ and variable pressure $p$, subject to no forces, and moving parallel to the complex plane is described by the equations¹

If $w=u-iv$ is an analytic function the components of the vector $\overline{w}=u+iv$ and

($p_{0}$ constant) satisfy these equations.

The function

determines a vector field. Find the direction and absolute value of $\overline{w}$ at the point $z=re^{i\vartheta }$, the potential, the conjugate potential, the level lines and the stream lines. (That part of the vector field that lies in the annulus $0<r_{1}<\left|z\right|<r_{2}$ can be considered as the electrostatic field between two condenser plates of a Leyden jar, or as the field of the heat flow in the chimney of a factory.)

Let $\varphi$ denote the potential and $\psi$ the conjugate potential of the vector field described in problem 91. Suppose that $z_{1}$ and $z_{2}$ are arbitrary points on the circles $\left|z\right|=r_{1}$ and $\left|z\right|=r_{2}$, respectively, and that $\varphi_{1}$ and $\varphi_{2}$ are the values of the potential at $z_{1}$ and $z_{2}$. Find

(difference of the potentials between the two condenser plates of the Leyden jar).

The conjugate potential $\psi$ in problem 91 turns out to be infinitely multivalued. Study the change in value along an arbitrary closed curve $L$ without double points that contains the circle $\left|z\right|=r_{1}$ and lies in the annulus $r_{1}<\left|z\right|<r_{2}$. For a given point $z$ of $L$ let $\psi$ be the value of the stream potential in $z$ before, and $\psi '$ be the value after, describing $L$ exactly once in the positive sense. Evaluate

(Flow of force passing from one condenser plate of the Leyden jar to the other.) Furthermore find

(Capacity of the cylindrical condenser plate per unit of the generatrix.)

Discuss the same questions as in problem 91 for the vector field determined by

(Stationary magnetic field of force generated by an infinite straight conductor perpendicular to the $z$-plane.) Are the potential $\varphi$ and the conjugate potential $\psi$ uniquely determined in this field?

Two infinite straight conductors are perpendicular to the $z$-plane piercing it at the points $z=-1$ and $z=1$. They carry currents of the same intensity but in opposite directions. Determine the stream lines and the level lines in the magnetic field so generated.

There are $n$ infinite straight conductors perpendicular to the $z$-plane piercing it at the points $z_{1},z_{2},\ldots ,z_{n}$. They carry currents of the same direction. There exist at most $n-1$ points in the $z$-plane where the generated magnetic force vanishes (points of equilibrium); these points lie in the smallest convex polygon that contains the points $z_{1},z_{2},\ldots ,z_{n}$.

Two confocal ellipses are given with foci at $z=-2$, $z=2$ and semi-axes $2a_{1},2b_{1}$ and $2a_{2},2b_{2}$

Find an irrotational and solenoidal vector field in the region bounded by the two ellipses such that these ellipses are level lines. (Electrostatic field in a condenser whose plates are confocal elliptic cylinders.) What do the stream lines and level lines look like? Find the capacity [problem 92].

Suppose $0<a<b$, $\alpha <\beta <\alpha +2\pi$. In the domain defined by the inequalities

determine an irrotational solenoidal vector field for which the bounding circular arcs are stream lines and the bounding straight segments are level lines (electric current in a plate of constant thickness).

We call $\psi_{1}$ and $\psi_{2}$ the values of the conjugate potential on $\left|z\right|=a$ and $\left|z\right|=b$ resp., $\varphi_{1}$ and $\varphi_{2}$ the values of the potential on $\arg z=\alpha$ and $\arg z=\beta$ resp. Compute

(Resistance, except for a factor depending on the thickness and the specific resistance of the plate.)

Find an irrotational solenoidal vector field outside the unit circle ($\left|z\right|\geq 1$); $\overline{w}$ ought to be $=1$ for $z=\infty$ and tangent to the unit circle. (A fluid passing a circular pillar; at a considerable distance off the pillar the flow is uniform.)

At which points of the vector field determined in problem 98 does the velocity vector vanish? (Stagnation points.) In how many points of the field does $\overline{w}$ assume the same value? Where is the pressure $p$ minimal, where is it maximal? [Problem 90.] What is the resultant pressure exerted on the pillar? Rotate all the vectors of the field through $90^{\circ }$: Give a physical interpretation of the vector field so obtained.

The figure represents the contours of a vector field determined by the following conditions: The irrotational and solenoidal field covers the entire upper, and part of the lower, half-plane; it is symmetric with respect to the imaginary axis. For $z=\infty$ the velocity $\overline{w}$ becomes $-i$, for $z=0$ we have $\overline{w}=0$. The following stream lines are known: the positive imaginary axis, the two pieces of real axis from $z=0$ to $z=l$ and from $z=0$ to $z=-l$ ($C$ to $A$, $A$ to $B$ and $A$ to $D$ in the figure); the direction of $\overline{w}$ is indicated by arrows.

The two remaining curvilinear parts of the boundary, which lead from $z=l$ and $z=-l$ ($B$ and $D$ in the figure) to $-i\infty$ should be determined so that they are at the same time stream lines and lines of constant velocity, i.e. $\overline{w}$ is tangent to, and $\left|\overline{w}\right|$ constant on, both curves. Draw the contours of the images of the field in the $\overline{w}$-plane and in the $f$-plane. (Stagnant water, ``dead water'', or infinite wake of constant pressure, behind a planklike barrier perpendicular to the direction of flow.) The segment from $z=-l$ to $z=l$ represents the barrier, the stagnation point $z=0$ is at its center. The undisturbed flow is assumed to be homogeneous with constant velocity $-i$. Throughout the wake the pressure is constant; this implies, according to Bernoulli's equation [problem 90], that $\left|\overline{w}\right| = \text{const.}$ along the stream line that separates the stagnant from the flowing water.

(continued). With the help of a conformal mapping express $w$ in terms of $f$ and then $z$ in terms of $f$ [problem 82]. Determine the width of the wake at a great distance.

(continued). Assuming that the density is $\varrho =1$, compute the total pressure against the barrier.