de_int

This is the numerical integration program using double exponential transformation. This code can calculate the integral from a to b and the 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.

$$I = \int_a^b f(x) \, dx$$ $$I = \int_a^\infty f(x) \, dx$$ $f(x)$ needs to be analytic within the integral region.  

Usage of the example code

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

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

Regarding the examples

• example.c

  • Example 1
    $$I = \int_a^b \frac{1}{\cos^2 x} \,dx = \tan b - \tan a $$
  • Example 2 $$I = \int_a^b \exp(ix) \,dx = -i(\exp(ib)-\exp(ia)) $$
  • Example 3 $$I(s) = \int_0^{\infty} \exp(-sx) \,dx = \frac{1}{s}, \ \ \ s > 0 $$
  • Example 4 $$I(s) = \int_0^{\infty} x \, \exp(-sx) \,dx = \frac{1}{s^2}, \ \ \ \Re s > 0 $$

  Examples 3 and 4 are known as Laplace transform related to the Heaviside step function.

• example2.c

  • Example 1 for straight line integral

    $$\begin{eqnarray} I &=& \int_C z^2 \, dz, \ \ \ C(t)=at+b, \, 0 \le t \le 1,\ &=& a \left( \frac{a^2}{3} + ab +b^2 \right). \end{eqnarray}$$

  • Example 2 for straight line integral

    $$\begin{eqnarray} I &=& \int_C \bar{z} \, dz, \ \ \ C(t)=at+b, \, 0 \le t \le t, \ &=& a \left( \frac{1}{2} \bar{a} + \bar{b} \right). \end{eqnarray}$$

  • Example 1 for contour integral

    $$\begin{eqnarray} I &=& \oint_C \frac{\sin z}{z^3 (z-2)} \, dz, \ \ \ C : |z-2|=1, \ &=& 2 \pi i \, \mathrm{Res}\left( \frac{\sin z}{z^3 (z-2)}, z=2 \right),\ &=& \frac{\pi i}{4} \sin 2. \end{eqnarray}$$

  • Example 2 for contour integral

    $$\begin{eqnarray} I &=& \oint_C \frac{\sin z}{z^3 (z-2)} \, dz, \ \ \ C : |z|=1, \ &=& 2 \pi i \, \mathrm{Res}\left( \frac{\sin z}{z^3 (z-2)}, z=0 \right),\ &=& -\frac{\pi i}{2}. \end{eqnarray}$$

Tips for improper integral

In the case of the upper or lower limit of the integration is a singular point, change the variables so that 0 is the singular point. This is to avoid a loss of significant digits near the singular point. When a is a singular point $$I = \int_a^b f(x) \, dx = \int_0^{b-a} f(t+a) \, dt, \ \ \ x=t+a.$$

Reference

Takahasi, Hidetosi, and Masatake Mori. "Double exponential formulas for numerical integration." Publications of the Research Institute for Mathematical Sciences 9.3 (1974): 721-741.

2022/08/05 Added examples of complex line integrals.