This is the numerical integration program for Fourier-type integrals, using double exponential transformation. This code can calculate the sine type integral and the cosine type integral from a to infinity shown by the following formulae. It also supports the integration of complex functions. I made it to integrate a complex function which has a singular point.

$$I = \int_a^{\infty} f(x) \sin(\omega x) \,dx $$

$$I = \int_a^{\infty} f(x) \cos(\omega x) \,dx $$

$f(x)$ needs to be analytic within the integral region, and non-oscillating function.

$$I = \int_a^{\infty} f(x) \,dx, $$

$$\qquad \qquad \qquad f(x) {\approx} g(x) \sin(\omega x), \ \ x{\gg}a. $$

$$I = \int_a^{\infty} f(x) \,dx, $$

$$\qquad \qquad \qquad f(x) {\approx} g(x) \cos(\omega x), \ \ x{\gg}a. $$

$g(x)$ needs to be analytic within the integral region, and non-oscillating function.

1. type 'make' command to compile.
2. type './example.out' to run.

Please see de_int_fi.h for detail of functions, example.c for detail of function usages.

• Example 1
  $$I = \int_0^{\infty} \frac{\sin x}{x} \,dx = \frac{\pi}{2} $$

• Example 2
  $$I = \int_0^{\infty} \frac{\cos x}{x^2+1} \,dx = \frac{\pi}{2e^{}} $$

  $e^{}$ is Napier's constant ( Euler's number ).

• Example 3
  $$I = \int_0^{\infty} \frac{\sin x}{x+1+i} \,dx = \mathrm{Ci}(z) \sin z + \left( \frac{\pi}{2} - \mathrm{Si}(z) \right) \cos z, \ \ z=1+i. $$

  $\mathrm{Ci}(z)$ is cosine integral function, $\mathrm{Si}(z)$ is sine integral function.

• Example 4
  $$I = \frac{2}{\pi i} \int_0^{\infty} \exp\left(-\frac{x-i}{\sqrt{2}}\right)\frac{\cos\left(\frac{\sqrt{x^2-2ix}}{\sqrt{2}}\right)}{\sqrt{x^2-2ix}} \,dx = H_0^{(1)}(1) $$

  $H_0^{(1)}(z)$ is Hankel function 1st-kind order 0.

Ooura, Takuya, and Masatake Mori. "A robust double exponential formula for Fourier-type integrals." Journal of computational and applied mathematics 112.1-2 (1999): 229-241.