Series and residues · Definition

Real integral by residues

complex analysisresidues

Before you start you need to know

It solves real integrals with no elementary antiderivative by moving them to the complex plane, where it is enough to count residues.

Definition

A method to compute \(\int_{-\infty}^{\infty} f(x)\,dx\) with \(f\) rational: close the real axis with a semicircle in the upper half-plane and apply the residue theorem.

\[\int_{-\infty}^{\infty} f(x)\,dx=2\pi i\sum_{\operatorname{Im}z_k>0}\operatorname{Res}(f,z_k)\]

Take the contour made of the segment \([-R,R]\) and the semicircle \(C_R\) of radius \(R\) in the upper half-plane. For large \(R\) it encloses every pole of \(f\) with positive imaginary part, and the residue theorem gives \(\int_{-R}^{R}f\,dx+\int_{C_R}f\,dz=2\pi i\sum\operatorname{Res}\).

Everything depends on the integral over \(C_R\) tending to \(0\). If \(f=P/Q\) with \(\deg Q\ge\deg P+2\), then \(|f(z)|\le M/R^2\) on \(C_R\) and the length \(\pi R\) gives the bound \(M\pi/R\to0\). With degree difference \(1\) the bound is not enough and another argument is needed. Also \(Q\) must not vanish on the real axis.

Guiding example: for \(\int_{-\infty}^{\infty}\frac{dx}{x^2+1}\) the only upper pole is \(i\), with residue \(\frac{1}{2i}\), and the value is \(2\pi i\cdot\frac{1}{2i}=\pi\). Typical mistakes: also adding the lower half-plane poles, or forgetting the factor \(2\pi i\).

One may also close in the lower half-plane, but then the contour is clockwise and each residue enters with a negative sign; the final value is the same.

Think about it

Why can't the formula be applied directly to \(\int_{-\infty}^{\infty}\frac{x}{x^2+1}\,dx\)?

Because \(\deg Q=\deg P+1\): the integral over \(C_R\) does not tend to \(0\) and the improper integral does not converge; it exists only as a principal value (\(0\), by oddness).

Compute \(\int_{-\infty}^{\infty}\frac{dx}{x^2+a^2}\) with \(a>0\).

The upper pole is \(ai\), with residue \(\frac{1}{2ai}\); the value is \(\pi/a\).

Exercises 10

Compute the following integrals

  1. \(\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+1}\)
  2. \(\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+4}\)
  3. \(\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+2x+2}\)
  4. \(\displaystyle \int_{-\infty}^{\infty}\frac{dx}{(x^{2}+1)^{2}}\)
  5. \(\displaystyle \int_{-\infty}^{\infty}\frac{dx}{(x^{2}+1)(x^{2}+4)}\)
  6. \(\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{4}+1}\)
  7. \(\displaystyle \int_{-\infty}^{\infty}\frac{dx}{(x^{2}+1)^{3}}\)

Problems

  1. For \(\int_{-\infty}^{\infty}\frac{dx}{x^{2}+9}\), which poles of \(1/(z^2+9)\) enter the sum of residues?
  2. Compute \(\operatorname{Res}\!\left(\frac{1}{z^{2}+1},\,i\right)\).
  3. Let \(f=P/Q\) be rational with no real poles. Which condition guarantees that the integral over the semicircle \(C_R\) tends to \(0\) as \(R\to\infty\)?

References

  • Elias M. Stein, Rami Shakarchi, Complex Analysis, 1st edition, Princeton University Press, 2003. Cap. 3, §2.1
  • James Ward Brown, Ruel V. Churchill, Complex Variables and Applications, 9th edition, McGraw-Hill, 2014. Cap. 7, §85
  • Lars V. Ahlfors, Complex Analysis, 3rd edition, McGraw-Hill, 1979. Cap. 4, §5.3