Recipe

Evaluate a contour integral

Use when: you're given \int_C f(z)\,dz for a specific contour C, stated as an explicit parametrization or as a curve you can parametrize yourself (a line segment, a circular arc, a bent path with corners)

Goal: the contour integral's value, as a single complex number, by rewriting it as an ordinary integral over the real parameter t

Taught in Contour integrals

1. 1

   Write down z(t) for a\le t\le b that traces C once. It must start and end at the stated points and move in the stated direction. A straight segment from z_0 to z_1 is z(t)=z_0+t(z_1-z_0), 0\le t\le1. A circular arc of radius r about c is z(t)=c+re^{it} over the angle range it sweeps.

   Example

   For the segment from 0 to 1+i, take z(t)=t+it, 0\le t\le1. Check the ends: z(0)=0 and z(1)=1+i. These match the start and end of the segment.

   • C is smooth, one continuous parametrization with no corners — continue straight through Steps 2 to 4 with that single z(t).
   • C is piecewise-smooth, built from segments or arcs joined at corners — parametrize each smooth piece on its own, then run Steps 2 through 4 on each piece before combining them in Step 5.

   Why: the definition of \int_C f(z)\,dz is stated with a parametrization. Without one, "integrate along the curve" is not yet an integral of anything.

2. 2

   Take the ordinary derivative of z(t) with respect to t. Use the same rules as in single-variable calculus. Just apply them to a complex-valued z(t).

   Example

   z(t)=t+it=(1+i)t, so z'(t)=1+i. It is a constant because z(t) is linear in t.

   Why: dz=z'(t)\,dt is the true complex step, with size and direction, along C. Skip it and you swap a walk along C for a walk along the plain t-axis.

3. 3

   Replace every z in f(z) with z(t) from Step 1. Multiply the result by z'(t) from Step 2. Then expand the algebra before you integrate.

   Example

   With f(z)=z, we get f(z(t))=z(t)=t+it. Multiply by z'(t)=1+i: (t+it)(1+i) = t+ti+ti+ti^2 = t+2ti-t = 2ti.

   Why: this is the integrand the definition \int_C f(z)\,dz=\int_a^b f(z(t))z'(t)\,dt calls for. Simplifying first keeps the real and imaginary parts from tangling once you integrate.

4. 4

   Integrate with respect to t over the stated interval. Treat i as a plain constant factor.

   Example

   \int_0^1 2ti\,dt = i\int_0^1 2t\,dt = i\big[t^2\big]_0^1 = i.

   Why: this is exactly the right side of the definition. It is an ordinary integral of a complex-valued function of the real variable t. Single-variable calculus already knows how to do that.

5. 5

   Say Step 1 split C into smooth pieces. Each piece now has its own number from Step 4. Add them all to get the total.

   Example

   Split a bent path into two segments and run Steps 2 through 4 on each. You might get \tfrac12 on the first and \tfrac12+i on the second. The integral over the whole path is their sum, \tfrac12+\left(\tfrac12+i\right)=1+i.

   Why: a contour integral over a piecewise-smooth curve is defined as the sum of the integrals over its smooth pieces. No shortcut skips computing each piece on its own.