Cauchy's Theorem

Evaluate

Evaluate

Let $a$ be a complex number with $\left|a\right| < 1$. Evaluate the integral

For $0 < a < b$, evaluate the integral

Prove the Fundamental Theorem of Algebra: Every nonconstant polynomial with complex coefficients has a complex root.

Let $f$ be continuous on $\mathbb {C}$ and analytic on $\left\{ z \mid \mathrm{Im}\left(z\right) \neq 0\right\}$. Prove that $f$ must be analytic on $\mathbb {C}$.

Let $f(z)$ be the rational function $p(z)/q(z)$, where $p(z)$ and $q(z)$ are nonzero polynomials with complex coefficients, such that the degree of $p(z)$ is less than the degree of $q(z)$, and such that $q(z)$ has no complex zeros with nonnegative imaginary part. Prove that if $z_{0}$ is a complex number with positive imaginary part, then

Let $f(z) = a_{0} + a_{1} z + \cdots + a_{n} z^{n}$ be a complex polynomial of degree $n > 0$. Prove

Let $f$ be an entire function such that

where $k$ is a positive integer and $A$ is a positive constant. Prove that $f$ is a constant multiple of the function $z^{k}$.

Let $f(z) = u(z) + iv(z)$ be holomorphic on $\left|z\right| < 1$, $u$ and $v$ real. Show that

for $0 < r < 1$ if $u(0)^{2} = v(0)^{2}$.

Let $f : \mathbb {C} \to \mathbb {C}$ be an analytic function such that

is bounded for some $k$ and $m$. Prove that $d^{n} f / dz^{n}$ is identically zero for sufficiently large $n$. How large must $n$ be, in terms of $k$ and $m$?

Suppose $\Omega$ is a bounded domain in $\mathbb {C}$ with a boundary consisting of a smooth Jordan curve $\gamma$. Let $f$ be holomorphic on a neighborhood of the closure of $\Omega$, and suppose that $f(z) \neq 0$ for $z \in \gamma$. Let $z_{1}, \ldots , z_{k}$ be the zeros of $f$ in $\Omega$, and let $n_{j}$ be the order of the zero of $f$ at $z_{j}$ (for $j = 1, \ldots , k$).

Let $f$ be an analytic function on a disc $D$ whose center is the point $z_{0}$. Assume that $\left|f'(z) - f'(z_{0})\right| < \left|f'(z_{0})\right|$ on $D$. Prove that $f$ is one-to-one on $D$.

Let $f(z)$ be analytic in the annulus $\Omega = \left\{ 1 < \left|z\right| < 2\right\}$. Assume that $f$ has no zeros in $\Omega$. Show that there exists an integer $n$ and an analytic function $g$ in $\Omega$ such that, for all $z \in \Omega$, $f(z) = z^{n} e^{g(z)}$.

Define the function $f$ on $\mathbb {C} \setminus [0,1]$ by

Prove that $f$ is analytic, and find its Laurent series about $\infty$.

Let the function $f$ be analytic and bounded in the complex half-plane $\mathrm{Re}\left(z\right) > 0$. Prove that for any positive real number $c$, the function $f$ is uniformly continuous in the half-plane $\mathrm{Re}\left(z\right) > c$.