Contour

A contour is a curve $C$ in the complex plane constructed by joining finitely many smooth curves end to end.

Riemann Sum

Let $C$ be a contour with initial point $A$ and terminal point $B$ .

A partition of $C$ is a finite set of points $P = \{z_0, z_1, \ldots, z_n\}$ , occurring in order along $C$ , with $z_0 = A$ and $z_n = B$ .

\Delta z_k = z_k - z_{k-1}, \qquad k = 1, 2, \ldots, n

$c_k$ is any point on the part of $C$ between $z_{k-1}$ and $z_k$ .

The Riemann sum for this partition, for a function $f$ is

\sum_{k=1}^n f(c_k)\Delta z_k

The contour integral of $f$ over $C$ is

\int_C f(z)\,\text{d}z = \lim_{\|P\| \to 0} \sum_{k=1}^n f(c_k)\Delta z_k

provided the limit exists and is independent of the choice of partition and points $c_k$ .

$\|P\|$ is the maximum length of the subarcs determined by $P$ .

Let $f(z)$ be continuous on a set containing the contour $C$ , and let $z = z(t)$ , $a \le t \le b$ , be a piecewise smooth parametrization of $C$ .

\int_C f(z)\,\text{d}z = \int_a^b f(z(t))z'(t)\,\text{d}t

When $z(t)$ is only piecewise smooth, the right side is evaluated piecewise.

Let $f(t) = u(t) + iv(t)$ be continuous on $[a,b]$ .

\left|\int_a^b f(t)\,\text{d}t\right| \le \int_a^b |f(t)|\,\text{d}t

Let $f$ be continuous on a contour $C$ , with $|f(z)| \le M$ for all $z$ on $C$ , and let $L$ be the length of $C$ .

\left|\int_C f(z)\,\text{d}z\right| \le ML

Let $C$ be the line segment from $0$ to $1+i$ , parametrized by $z(t) = t(1+i)$ , $0 \le t \le 1$ . $z'(t) = 1+i$ .

\int_C z\,\text{d}z = \int_0^1 t(1+i)(1+i)\,\text{d}t = (1+i)^2\int_0^1 t\,\text{d}t

$(1+i)^2 = 2i$ and $\displaystyle\int_0^1 t\,\text{d}t = \dfrac{1}{2}$

\int_C z\,\text{d}z = 2i \cdot \frac{1}{2} = i