Cauchy's Formula

Put

Compute the sum

which converges to the integral $\oint \dfrac {dz}{z}$ along $\left|z\right|=\left|a\right|$ as $n\to \infty$.

Let $k$ denote an integer different from $-1$ and $L$ a closed curve without double points and of finite length; if $k\leq -2$, $L$ does not pass through $z=0$; furthermore the points $z_{1},z_{2},\ldots ,z_{n}$ are consecutive points on $L$. Show that the integral $\oint_{L} z^{k}\, dz$ vanishes; try to approximate it by a sum of the form

Evaluate

along the ellipse $z=x+iy$, $x^{2}-xy+y^{2}+x+y=0$.

Show that

Prove the formula

Evaluate the curvilinear integral

along an ellipse with the foci $(0,-1)$ and $(0,1)$.

We have for $0<\mathrm{Re}\left(s\right)<1$

The equation

holds for $n>1$.

We find, assuming $\mu >0$, $0<\alpha <\frac{\pi }{2}$, $n=0,1,2,\ldots$, that

Note the special case $\alpha =\mu \pi$.

Assume $\mu >0$, $x>0$, $\mu$ fixed, $x$ variable. Show that

Let $a>0$. The integral

along the straight line $s=a+it$, $-\infty <t<\infty$, parallel to the imaginary axis converges absolutely for all real values of $\alpha$. The integral turns out to be

We call $\mu (t)$ the largest term of the series

Assume $\lambda >0$ and let $z$ be the only positive root of the equation

Then

The Legendre polynomials can be defined as the coefficients of the expansion into a power series [Part VI, problem 91]

Deduce from this Laplace's formula (Part VI, problem 86)

and the Dirichlet-Mehler formula

(The square roots are positive.)

Let $\mathfrak {H}$ denote the half-strip

and $L$ the boundary of $\mathfrak {H}$ consisting of three straight pieces. We orient $L$ so that $\mathfrak {H}$ is on the right hand side of $L$. The integral

defines a function $E(z)$ for points $z$ on the left hand side of $L$. Show that $E(z)$ is an entire function which assumes real values for real $z$.

(continued). We find

(continued). The function

is bounded outside $\mathfrak {H}$. Inside $\mathfrak {H}$ the function

is bounded.

We define

Show that

where the integrals are computed along the positively oriented circles. [Part II, problem 217.]

Assume that $f(z)$ is regular in the disk $\left|z\right|\leq r$ and different from $0$ on the circle $\left|z\right|=r$. The largest value of $\re z\dfrac {f’(z)}{f(z)}$ on $\left|z\right|=r$ is at least equal to the number of zeros of $f(z)$ in $\left|z\right|<r$.

Let $z_{1},z_{2},\ldots ,z_{n}$ be arbitrary but distinct complex numbers and $L$ be a closed continuous curve without double points, enclosing all the $z_{\nu }$'s. The function $f(z)$ is supposed to be regular inside of, and on, $L$. Then

where $\omega (z)=(z-z_{1})(z-z_{2})\cdots (z-z_{n})$, is the uniquely determined polynomial of degree $n-1$ that coincides with $f(z)$ at the points $z_{1},z_{2},\ldots ,z_{n}$.

The function $f(z)$ is analytic on the segment $a\leq z\leq b$ of the real axis and it assumes there real values. The closed curve $L$ is continuous without double points and encloses the segment $a\leq z\leq b$; $f(z)$ is regular inside $L$. Let $z_{1},z_{2},\ldots ,z_{n}$ denote arbitrary points of the real segment $[a,b]$. Then there exists a point $z_{0}$ on $[a,b]$ such that

Assume that the entire function $F(z)$ satisfies the inequality

in the entire $z$-plane, $z=x+iy$; $C$ and $\varrho$ are positive constants. Then

Example: $F(z)=\cos \varrho z$.

The entire function $G(z)$ is supposed to satisfy the inequality

on the entire $z$-plane, $z=x+iy$; $C,\varrho$ positive constants; in addition assume that $G(z)$ is an odd function, $G(-z)=-G(z)$. Then

Example: $G(z)=\sin \varrho z$.

We suppose that the function $f(z)$ is regular in the disk $\left|z\right|\leq 1$. Then

We integrate with respect to $x$ along a straight line from $0$ to $1$, with respect to $z$ along the positively oriented unit circle. We start at the point $z=1$ with the branch of $\log z$ that is real, and the branch of $z^{k}$ that is positive, for positive $z$.

The function $f(z)$ is regular in the unit disk $\left|z\right|\leq 1$ and satisfies the condition

Let $k>-1$. Then

Let $\alpha >-2$. The quadratic form of the infinitely many real variables $x_{1},x_{2},x_{3},\ldots$

is bounded, i.e.there exists a constant $M$ independent of $n$ such that

whenever the variables $x_{1},x_{2},\ldots ,x_{n}$ satisfy the condition $x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=1$; $n=1,2,3,\ldots$ We may choose $M=\pi$ if $\alpha$ is an integer and $M=\dfrac {\pi }{\left|\sin \alpha \pi \right|}$ if $\alpha$ is not an integer.

We assume that the functions $f_{1}(z),f_{2}(z),\ldots ,f_{n}(z),\ldots$ are regular in the open region $\mathfrak {R}$ and that they converge uniformly to the function $f(z)$ in any closed domain inside $\mathfrak {R}$. Then the limit function $f(z)$ is regular in $\mathfrak {R}$.

The complex function $f(z)=u(x,y)+iv(x,y)$ of the real variables $x$ and $y$ is defined and continuous in a region $\mathfrak {R}$ of the $z=x+iy$ plane. Moreover we assume that the integral

along any circle inside $\mathfrak {R}$ vanishes. Then $f(z)$ is an analytic function of the complex variable $z$, regular in the entire region $\mathfrak {R}$.