Complex integration · Definition

Contour integral

complex analysiscomplex integration

Before you start you need to know

It is the way to add up a complex function along a path in the plane; from it come Cauchy's theorem, residues and nearly all the calculation of real integrals using complex variables.

Definition

The contour integral of a continuous function \(f\) over a contour \(C\) with parametrization \(\gamma\) is the limit of sums of \(f\) evaluated at points of \(C\) times the small complex displacements.

\[\int_C f(z)\,dz=\int_a^b f(\gamma(t))\,\gamma'(t)\,dt\]

In practice the integral is computed with the parametrization: if \(C\) has parametrization \(\gamma:[a,b]\to\mathbb{C}\), then \(\int_C f\,dz=\int_a^b f(\gamma(t))\gamma'(t)\,dt\). The integrand is a complex function of the real variable \(t\), and real and imaginary parts are integrated separately. For the segment from \(0\) to \(1+i\), with \(\gamma(t)=(1+i)t\), one gets \(\int_C z^2\,dz=(1+i)^3/3\).

The integral is linear in \(f\), additive over the pieces of a contour and changes sign when the direction is reversed: \(\int_{-C}f\,dz=-\int_Cf\,dz\). It does not depend on the chosen parametrization as long as it traverses the curve in the same direction. It also satisfies the bound \(\left|\int_Cf\,dz\right|\le ML\), with \(M\) the maximum of \(|f|\) on \(C\) and \(L\) the length of \(C\).

The result may depend on the path: \(\int_C\bar z\,dz\) from \(0\) to \(1+i\) along the segment equals \(1\), but along the path that first goes to \(1\) and then up it equals \(1+i\). By contrast, for \(f(z)=z^2\) the value depends only on the endpoints. Understanding when each case happens is the content of Cauchy's theorem.

The central example is \(\oint_{|z|=1}z^n\,dz\) counterclockwise: it equals \(2\pi i\) if \(n=-1\) and \(0\) for any other integer. That integral, \(\oint dz/z=2\pi i\), is the axis of Cauchy's integral formula and of residues.

Think about it

What is \(\int_C dz\) over any contour from \(z_0\) to \(z_1\)?

\(z_1-z_0\): with \(f=1\), \(\int_a^b\gamma'(t)\,dt=\gamma(b)-\gamma(a)\).

Why does \(\oint_{|z|=1}\bar z\,dz\) not vanish if \(\oint z\,dz\) does?

Because \(z\) is holomorphic and \(\bar z\) is not; for holomorphic functions the integral over a closed contour vanishes (Cauchy's theorem), whereas \(\bar z\) gives \(2\pi i\).

Exercises 10

  1. Compute \(\displaystyle\int_C z^2\,dz\), with \(C\) the segment from \(0\) to \(1+i\).
  2. Compute \(\displaystyle\oint_{|z|=1}\frac{dz}{z}\), with the circle traversed once counterclockwise.
  3. Compute \(\displaystyle\oint_{|z|=1}z\,dz\) counterclockwise.
  4. Compute \(\displaystyle\oint_{|z|=1}\bar z\,dz\) counterclockwise.
  5. Compute \(\displaystyle\int_C\bar z\,dz\), with \(C\) the segment from \(0\) to \(1+i\).
  6. If \(\displaystyle\int_C f\,dz=3-i\), what is the integral over the opposite contour \(-C\)?
  7. A contour \(C\) is made of \(C_1\) and \(C_2\). If \(\int_{C_1}f\,dz=2i\) and \(\int_{C_2}f\,dz=1-i\), what is \(\int_C f\,dz\)?
  8. Compute \(\displaystyle\int_C\operatorname{Re}(z)\,dz\), with \(C\) the segment from \(0\) to \(1+i\).
  9. Let \(C\) be the circle \(|z|=2\) traversed once. Use \(\left|\int_C f\,dz\right|\le ML\) with \(f(z)=1/z^2\) to bound \(\left|\int_C f\,dz\right|\).
  10. For an integer \(n\), for which value is \(\displaystyle\oint_{|z|=1}z^n\,dz\) (counterclockwise) nonzero?

What comes next

References

  • Elias M. Stein, Rami Shakarchi, Complex Analysis, 1st edition, Princeton University Press, 2003. Cap. 1, §2.2
  • James Ward Brown, Ruel V. Churchill, Complex Variables and Applications, 9th edition, McGraw-Hill, 2014. §40–43
  • Lars V. Ahlfors, Complex Analysis, 3rd edition, McGraw-Hill, 1979. Cap. 4, §1.2