Math Problems Vault

Problem 5.1

Find all points $z = x+i y$ at which the following functions are differentiable:

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(1)

$\mathrm{Im}\left(z\right)$ ;

(2)

$\left|\bar{z}\right|^{2}$ ;

(3)

$x^{2}-i y^{2}$ ;

(4)

$x^{2}-y^{2}-2 i x y$ ;

(5)

$x-y+i(x+y)$ ;

(6)

$z \mathrm{Re}\left(z\right)$ .

Problem 5.2

Prove that for any natural value $n$ the function $z^{n}$ is differentiable in the entire complex plane and that $\left(z^{n}\right)^{\prime } = n z^{n-1}$ .

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Problem 5.3

Prove that every polynomial in $z$ is a differentiable function in the entire complex plane, and every rational function is differentiable at every point where the denominator does not vanish.

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Problem 5.4

Define the function $e^{z}$ for any complex value $z = x+i y$ by the equality $e^{z} = e^{x}(\cos y+i \sin y)$ . Prove that the function $e^{z}$ is differentiable for any $z$ and the equality $\left(e^{z}\right)^{\prime } = e^{z}$ holds.

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Problem 5.5

(1)(2)(3)(4)

Define the functions $\operatorname {sh} z, \operatorname {ch} z, \sin z, \cos z$ by the equalities

\begin{aligned} \operatorname {sh} z & = \frac{1}{2}\left(e^{z}-e^{-z}\right), & \operatorname {ch} z & = \frac{1}{2}\left(e^{z}+e^{-z}\right), \\ \sin z & = \frac{1}{2 i}\left(e^{i z}-e^{-i z}\right), & \cos z & = \frac{1}{2}\left(e^{i z}+e^{-i z}\right). \end{aligned}

Prove that

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(1)

$(\operatorname {sh} z)^{\prime } = \operatorname {ch} z$ ;

(2)

$(\operatorname {ch} z)^{\prime } = \operatorname {sh} z$ ;

(3)

$(\sin z)^{\prime } = \cos z$ ;

(4)

$(\cos z)^{\prime } = -\sin z$ .

Problem 5.6

Determine at which points $z \in \mathbb {C}$ the functions are differentiable, and find their derivatives:

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(1)

$e^{\operatorname {sh} z}$ ;

(2)

$\cos \left(2 e^{z}\right)$ ;

(3)

$\sin z \operatorname {sh} z+i \cos z \operatorname {ch} z$ ;

(4)

$\frac{e^{z}}{z}$ ;

(5)

$\frac{z}{e^{z}}$ ;

(6)

$\frac{\sin z}{1+z^{2}}$ .

Problem 5.7

Determine where the functions are differentiable, and find their derivatives:

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(1)

$\operatorname {tg} z$ ;

(2)

$\operatorname {ctg} z$ ;

(3)

$\frac{e^{z}+2}{e^{z}-2}$ ;

(4)

$\frac{1}{\operatorname {tg} z+\operatorname {ctg} z}$ ;

(5)

$\left(e^{z}+e^{-z}\right)^{-3}$ ;

(6)

$\frac{\sin z}{\sin z-\cos z}$ .

Problem 5.8

(1)(2)(3)

Let the function

f(z) = u(x, y)+i v(x, y)

be differentiable at the point $z_{0} = x_{0}+i y_{0}$ . Prove the validity of the formulas:

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(1)

$f^{\prime }\left(z_{0}\right) = \frac{\partial v}{\partial y}\left(x_{0}, y_{0}\right)-i \frac{\partial u}{\partial y}\left(x_{0}, y_{0}\right)$ ;

(2)

$f^{\prime }\left(z_{0}\right) = \frac{\partial u}{\partial x}\left(x_{0}, y_{0}\right)-i \frac{\partial u}{\partial y}\left(x_{0}, y_{0}\right)$ ;

(3)

$f^{\prime }\left(z_{0}\right) = \frac{\partial v}{\partial x}\left(x_{0}, y_{0}\right)+i \frac{\partial v}{\partial x}\left(x_{0}, y_{0}\right)$ .

Problem 5.9

(1)(2)(3)(4)

Let the function $f(z) = u(x, y)+i v(x, y)$ be differentiable in a domain $G \subset \mathbb {C}$ , and let one of the functions

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(1)

$u(x, y)$ ;

(2)

$v(x, y)$ ;

(3)

$r(x, y) = \left|f(z)\right|$ ;

(4)

$\varphi (x, y) = \arg f(z)$ retain a constant value in the domain $G$ . Prove that $f(z) \equiv$ const.

Problem 5.10

Let $u, v$ be a pair of conjugate harmonic functions in a domain $G$ , and let $\xi , \eta$ be a pair of conjugate harmonic functions in a domain $D$ . Let, for every $z = x+i y \in G$ , the value $u(x, y)+i v(x, y)$ belong to the domain $D$ . Prove that the pair of functions $U, V$ of the form

\begin{aligned} U(x, y) & = \xi (u(x, y), v(x, y)) \\ V(x, y) & = \eta (u(x, y), v(x, y)) \end{aligned}

forms a pair of conjugate harmonic functions in the domain $G$ .

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Problem 5.11

Let $u, v$ be a pair of conjugate harmonic functions in a domain $G$ , and let the functions $u$ and $v$ not vanish simultaneously at any point of the domain $G$ . Prove that the function

U(x, y) = \ln \left[u^{2}(x, y)+v^{2}(x, y)\right]

is harmonic in the domain $G$ .

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Problem 5.12

Let $u, v_{1}$ and $u, v_{2}$ be two pairs of conjugate harmonic functions in a domain $G$ with the same first function $u$ . Prove that $v_{1}(x, y)-v_{2}(x, y) \equiv$ const.

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Problem 5.13

(1)(2)(3)(4)

In the following problems, one of the pair of conjugate harmonic functions $u$ or $v$ is given. Find the second function of the pair.

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(1)

$u = x y$ ;

(2)

$u = x^{2}-y^{2}+2 x y$ ;

(3)

$v = y \sin x \operatorname {ch} y+x \operatorname {sh} y \cos x$ ;

(4)

$u = r \varphi \cos \varphi +r \ln r \sin \varphi \quad \left(z = x+i y = r e^{i \varphi }\right)$ .

Problem 5.14

(1)(2)(3)(4)

Find all harmonic functions of the form

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(1)

$u = \varphi \left(x^{2}+y^{2}\right)$ ;

(2)

$u = \varphi \left(x^{2}-y^{2}\right)$ ;

(3)

$u = \varphi \left(\frac{y}{x}\right)$ ;

(4)

$u = \varphi \left(\frac{x^{2}+y^{2}}{x}\right)$ .

Problem 5.15

(1)(2)

Let $P(x, y)$ be a polynomial in $x$ and $y$ with complex coefficients. Denote

P^{*}(z, \bar{z}) = P\left(\frac{z+\bar{z}}{2}, \frac{z-\bar{z}}{2 i}\right)

Prove that

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(1)

the functions $u(x, y) = \mathrm{Re}\left(P(x, y)\right), v(x, y) = \mathrm{Im}\left(P(x, y)\right)$ satisfy the Cauchy--Riemann conditions if and only if the polynomial $P^{*}(z, \bar{z})$ does not depend on $\bar{z} ;$

(2)

the polynomial $P(x, y)$ satisfies Laplace's equation $\Delta P = 0$ if and only if the polynomial $P^{*}(z, \bar{z})$ can be represented in the form $Q_{1}(z)+Q_{2}(\bar{z})$ , where $Q_{1}$ and $Q_{2}$ are polynomials.

Problem 5.16

(1)(2)

Let $P(z)$ be a polynomial. Set

\begin{aligned} & u(x, y) = \mathrm{Re}\left(P(x+i y)\right) ; \\ & v(x, y) = \mathrm{Im}\left(P(x+i y)\right). \end{aligned}

Prove that the formulas hold

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(1)

$P(z) = 2 u\left(\frac{z}{2}, \frac{z}{2 i}\right)-\overline{P(0)}$ ;

(2)

$P(z) = 2 i v\left(\frac{z}{2}, \frac{z}{2 i}\right)+\overline{P(0)}$ .

Problem 5.17

Recover the regular function $f(z)$ from the condition

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(1)

$\mathrm{Re}\left(f(z)\right) = x^{3}-6 x^{2} y-3 x y^{2}+2 y^{3}, f(0) = i$ ;

(2)

$\mathrm{Re}\left(f(z)\right) = x \sin x \operatorname {ch} y-y \operatorname {sh} y \cos x, \quad f(0) = 0$ ;

(3)

$\mathrm{Im}\left(f(z)\right) = y \operatorname {ch} x \cos y+x \sin y \operatorname {sh} x, f(0) = 1$ ;

(4)

$\mathrm{Im}\left(f(z)\right) = y \cos x \operatorname {ch} y-x \sin x \operatorname {sh} y, f(0) = 2$ ;

(5)

$\mathrm{Im}\left(f(z)\right) = x \sin y \operatorname {ch} x+y \operatorname {sh} x \cos y, f(0) = 0$ ;

(6)

$\mathrm{Re}\left(f(z)\right) = x e^{x} \cos y-(y+1) e^{x} \sin y, \quad f(0) = i$ ;

(7)

$\left|f(z)\right| = \left(x^{2}+y^{2}\right) e^{x}$ ;

(8)

$\arg f(z) = 2 x y$ ;

(9)

$\left|f(z)\right| = r e^{r^{2} \cos 2 \varphi } \quad \left(z = r e^{i \varphi }\right)$ ;

(10)

$\arg f(z) = \varphi +r \sin \varphi \left(z = r e^{i \varphi }\right)$ .

Problem 5.18

Let the functions $f(z)$ and $g(z)$ be regular in some domain $G$ . Prove that the sum $f(z)+\overline{g(z)}$ is real throughout the domain $G$ if and only if $f(z) = g(z)+C$ , where $C$ is a real constant.

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Problem 5.19

Let the functions $f(z)$ and $g(z)$ be regular in some domain $G$ , and $g(z) \not\equiv 0$ . Prove that the product $f(z) \overline{g(z)}$ is real (nonnegative) throughout the domain $G$ if and only if $f(z) = C g(z)$ , where $C$ is a real (nonnegative) constant.

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