Regular Functions

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

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

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.

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.

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

Prove that

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

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

Let the function

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

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

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

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

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

is harmonic in the domain $G$.

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.

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

Find all harmonic functions of the form

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

Prove that

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

Prove that the formulas hold

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

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.

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.

Prove that if the function $f(z)$ is regular in the disk $\left|z-a\right| < R$ and satisfies the condition $\left|f(z)\right| \leqslant M \quad (\left|z-a\right| < R)$, then for any two points $z_{1}$ and $z_{2}$ of this disk the inequality holds

Prove that the assertion of Problem 1 remains valid if the domain of regularity of the function $f(z)$ is not necessarily a disk, but an arbitrary convex domain in $\mathbb {C}$.

Let the function $f(z)$ be regular in a convex domain $D \subset \mathbb {C}$ and satisfy the condition $\mathrm{Re}\left(f(z)\right) \geqslant M > 0 \quad (z \in D)$. Prove that for any two points $z_{1}$ and $z_{2}$ of this domain

Prove that the assertion of Problem 3 remains valid if the condition

is replaced by the condition $\mathrm{Re}\left(\left\{ e^{i \varphi } f(z)\right\} \right) \geqslant M$ (the real number $\varphi$ does not depend on the point $z$).

Let the function $f(z)$ be regular in a simply connected domain $D \subset \mathbb {C}$. Prove that the function

is an antiderivative of the function $f(z)$.

Let the functions $f(z)$ and $g(z)$ be regular in a simply connected domain $D \subset \mathbb {C}$, and let $F(z)$ and $G(z)$ be, respectively, antiderivatives of these functions. Prove the integration-by-parts formula:

Find the antiderivatives of the functions:

Let the function $f(z)$ be regular in the annulus $r < \left|z-a\right| < R$. Prove that the integral

does not depend on the number $\rho$ (the circle is traversed counterclockwise).

Let the function $f(z)$ be regular in the annulus $r < \left|z\right| < R$, and let a simple piecewise-smooth curve $\gamma$ bound a domain that contains the disk $\left|z\right| \leqslant r$ and lies in the disk $\left|z\right| < R$, so that while traversing the curve $\gamma$ the domain remains on the left. Prove that the integral $\int_{\gamma } f(z) d z$ does not depend on the choice of the curve $\gamma$ (satisfying the conditions stated above).

Let the function $f(z)$ be regular in an arbitrary domain $D$ of the complex plane. Prove that a necessary and sufficient condition for the existence of an antiderivative of the function $f(z)$ in the domain $D$ is that the integral of the function $f(z)$ over any simple closed polygonal line lying in the domain $D$ be equal to zero.

Prove that the following functions have no antiderivatives in the domains indicated in parentheses:

Let the function $f(z)$ be regular in a bounded doubly connected domain $D$, enclosed between two closed piecewise-smooth curves $\gamma_{1}$ and $\gamma_{2}$, and continuous up to its boundary. Prove that the function $f(z)$ has an antiderivative in the domain $D$ if and only if

Let the function $f(z)$ be regular in a domain, simply connected in $\overline{\mathbb {C}}$, containing the point $z = \infty$. Prove that the function $f(z)$ has an antiderivative in this domain if and only if $\lim_{z \rightarrow \infty } z f(z) = 0$.

Let the function $f(z)$ be regular in a simply connected domain $D \subset \overline{\mathbb {C}}$ containing the point $z = \infty$. Denote by $D^{\prime }$ the domain $D$ with the point $z = \infty$ removed. Prove that the function $f(z)$ has an antiderivative in the domain $D^{\prime }$ if and only if

Let the functions $f(z)$ and $g(z)$ be regular in a domain $D \subset \overline{\mathbb {C}}$ containing the point $z = \infty$, and have antiderivatives in the domain $D^{\prime }$ (the domain $D$ with the point $z = \infty$ removed). Prove that the functions $f(z)+g(z), f(z) g(z), P(f(z)), e^{f(z)} (P(w)$ an arbitrary polynomial) are also regular in the domain $D$ and have an antiderivative in the domain $D^{\prime }$.

Let the function $f(z)$ be regular in a bounded $m$-connected domain $D$ whose boundary consists of closed piecewise-smooth curves $\Gamma_{1}, \Gamma_{2}, \ldots , \Gamma_{m}$. Prove that for the existence of an antiderivative of the function $f(z)$ in the domain $D$ it is necessary and sufficient that

Let the function $f(z)$ be regular in the strip $-a < \mathrm{Im}\left(z\right) < a$ and satisfy the condition

Prove that if the integral $\int_{-\infty }^{\infty } f(x) d x$ converges, then for any $\alpha$ in the interval $(-a, a)$ the integral $\int_{i \alpha -\infty }^{i \alpha +\infty } f(z) d z$ also converges and does not depend on $\alpha$.

Let the function $f(z)$ be regular in the angle $-a < \arg z < a$ and satisfy the conditions

Prove that if the integral $\int_{0}^{\infty } f(x) d x$ converges, then for any $\alpha$ in the interval $(-a, a)$ the integral $\underset {\arg z = \alpha }{\int } f(z) d z$ also converges and does not depend on $\alpha$.

It is known from analysis that $\int_{-\infty }^{\infty } e^{-x^{2}} d x = \sqrt{\pi }$. Compute the integral

Compute the integral $\int_{0}^{+\infty } e^{-i x^{2}} d x$.

Using Cauchy's integral formula, compute the integrals (all circles are traversed counterclockwise):

Let the functions $f(z)$ and $g(z)$ be regular in the disk $\left|z\right| < 1$ and continuous in the closed disk $\left|z\right| \leqslant 1$. Prove that

Let the function $f(z)$ be regular in a domain $D \subset \overline{\mathbb {C}}$ containing the point $z = \infty$, and continuous up to its boundary. Prove that in this case Cauchy's integral formula takes the form

and the formula for the derivatives of $f(z)$ retains its previous form.

Let the function $f(z)$ be regular in the disk $\left|z-a\right| < R$ and continuous in the closed disk $\left|z-a\right| \leqslant R$. Prove the formula

known as the mean value theorem.

Let the function $f(z)$ be regular in the disk $\left|z\right| < R$ and continuous in the closed disk $\left|z\right| \leqslant R$. Compute the integral

Prove that a function, regular in some domain and not identically constant, cannot attain its greatest absolute value at an interior point of this domain. (The maximum modulus principle.)

Let the function $u(x, y)$ be harmonic in the disk $\left|z-a\right| \leqslant R$. Prove that

Prove that a function, harmonic in some domain and not identically constant, can attain neither its greatest nor its least value at an interior point of this domain. (The maximum principle for harmonic functions.)

Let the function $f(z)$ be regular in a bounded domain $D \subset \mathbb {C}$ and continuous up to its boundary $\partial D$, consisting of a finite number of closed piecewise-smooth curves. Prove the inequality

where $M = \max_{z \in \partial D}\left|f(z)\right|, \rho$ is the distance from the point $z$ to the boundary of the domain $D$, and $L$ is the total length of the boundary of the domain $D$.

Let the function $f(z)$ be regular in the disk $\left|z\right| < R$ and continuous in the closed disk $\left|z\right| \leqslant R$. Prove the inequality

where $M = \max_{\left|z\right| = R}\left|f(z)\right|$.

Let the function $f(z)$ be regular in the disk $\left|z\right| < R$ and continuous in the closed disk $\left|z\right| \leqslant R$. Prove the inequality

where $M = \max_{\left|z\right| = R}\left|f(z)\right|$, and show that for $n = 1$ this inequality can be written in the form

Let the function $f(z)$ be regular in the entire plane and satisfy the condition $\left|f(z)\right| \leqslant M$ for all $z$. Prove that $f(z)$ is identically constant. (Liouville's theorem.)

Let the function $f(z)$ be regular in the entire plane and satisfy the inequality

Prove that $f(z)$ is a polynomial of degree not exceeding $p$.

At all points not belonging to the curve of integration, compute the values of the following Cauchy-type integrals:

Let the function $f(z)$ be regular in the annulus $r < \left|z-a\right| < R$. Prove that the Cauchy-type integral

does not depend on the number $\rho$ for $\left|z\right| < r$ (or for $\left|z\right| > R$).

Let the function $f(\zeta )$ be regular in the annulus $r < \left|\zeta \right| < R$ and continuous in the closed annulus $r \leqslant \left|\zeta \right| \leqslant R$. Denote

(the circles are traversed counterclockwise). Prove that for $r < \left|z\right| < R$ the equality $f(z) = f_{1}(z)+f_{2}(z)$ holds.

Let the function $f(z)$ be regular in the annulus $r < \left|z-a\right| < R$. Prove that it can be represented in the form $f(z) = f_{1}(z)+f_{2}(z)$, where the function $f_{1}(z)$ is regular in the disk $\left|z-a\right| < R$, and the function $f_{2}(z)$ is regular for $\left|z-a\right| > r$ and $f_{2}(\infty ) = 0$.

Prove that the representation of the function $f(z)$ from the previous problem is unique.

Let a simple closed curve $\gamma_{1}$ bound a domain $D_{1} \subset \mathbb {C}$, and let a simple closed curve $\gamma_{2}$, lying in the domain $D_{1}$, bound a domain $D_{2} \subset D_{1}$. Prove that any function $f(z)$, regular in the ring-shaped domain $D = D_{1} \backslash \bar{D}_{2}$, can be represented in the form $f(z) = f_{1}(z)+f_{2}(z)$, where the function $f_{1}(z)$ is regular in the domain $D_{1}$, and the function $f_{2}(z)$ is regular outside the domain $D_{2}$. Prove also that such a representation is unique if the condition $f_{2}(\infty ) = 0$ is imposed.

Let $\gamma$ be a simple closed curve bounding a domain $D \subset \mathbb {C}$, and let the function $\varphi (z)$ be regular in some domain containing the curve $\gamma$. Prove that for the existence of a function $f(z)$, regular in the closure of the domain $D$ and coinciding with the function $\varphi (z)$ on the curve $\gamma$, it is necessary and sufficient that

Let $\gamma$ be a simple closed piecewise-smooth curve bounding a bounded domain $D$, and let the function $\varphi (t)$ be continuous on the curve $\gamma$. Prove that for the equality

to hold it is necessary and sufficient that

and for the equality

to hold it is necessary and sufficient that