Numerical integration over the infinite domain (-∞, ∞)
Compute the integral over the infinite domain (-∞, ∞) using the double exponential (DE) numerical integration method. This is suitable when the function decays algebraically at infinity.
The integrand f(x) is assumed to be analytic except at the endpoints and not to be a periodic function.
| \(\normalsize Double\ exponential\ integration\\ (1)\ x\rightarrow \sinh(\frac{\pi}{2}\sinh(t))\\ \hspace{10px} {\large\int_{\small -\infty}^{\small\infty}}f(x)dx={\large\int_{\small -\infty}^{\small\infty}}f(x(t))x'(t)dt\\ \hspace{100px}x(t)=\sinh(\frac{\pi}{2}\sinh(t))\\ \hspace{100px}x'(t)=\cosh(\frac{\pi}{2}\sinh(t))\frac{\pi}{2}\cosh(t)\\ (2)\ Trapezoid\\ \hspace{20px}S\simeq{\large\int_{-t_a}^{t_a}}f(x(t))x'(t)dt=h{\large\displaystyle \sum_{{\small j=-}\frac{N}{2}}^{\frac{N}{2}}}f(x(jh))x'(jh)\\ \hspace{240px}h={\large\frac{2t_a}{N}}\\ \) |
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