A typical undergraduate student (at least in North America) learns about integration of real-valued functions of one real variable, and learns some of its applications to science and probability, e.g. computing things like mass, charge, work, or the expected value of a continuous random variable. They are also taught several techniques of integration such as change of variables, integration by parts, partial fraction decomposition, etc. At some point later one, they learn some complex analysis, and in particular the Residue Theorem. With some ingenuity, one can use this theorem about integrating complex functions over curves in the complex plane to compute integrals of real functions over (parts of) the real line, e.g.:
$$\int_0^\infty\frac{\cos(ax)}{x^2+1}dx\ \ \ \ \ \ \ \ \ \ \ (a > 0)$$
My question is as in the title. For instance, has there been some point in history when physicists benefited greatly from knowing how to calculate a certain real integral, whose computation via non-complex methods was not known at the time, or was perhaps overly difficult?
Best Answer
It's used all the time in physics, especially in quantum mechanics. The math physics text book "Mathematical Methods for Physicists" by Arfken & Weber has three applications in chapter 7.1 in the examples and problems:
Forced classical oscillator: $\ddot{x} + \omega_0x(t) = f(t),$ where you find the Green's function.
Quantum Mechanical scattering: $I(\sigma) = \int_{-\infty}^{\infty} \frac{x \sin(x)\;dx}{x^2-\sigma^2}$
Quantum theory of atomic collisions: $I = \int_{-\infty}^{\infty} \frac{\sin(t)}{t}\;e^{ipt}\;dt$