Numerical integration over the finite interval (a, b)
Computes the integral over the finite interval (a, b) using the double exponential (DE) numerical integration method. It is also suitable for integrating functions whose endpoints are singular points (±∞).
The integrand f(x) is assumed to be analytic except at the endpoints and not to be a periodic function.
| \( \normalsize Double\ exponential\ integration\\ (1)\ x\rightarrow \frac{b-a}{2}y+\frac{b+a}{2}\\ \hspace{30px} {\large\int_{\small a}^{\small b}}f(x)dx= {\large\int_{\small -1}^{\small 1}}f(\frac{b-a}{2}x+\frac{b+a}{2})\frac{b-a}{2}dx\\ (2) x\rightarrow \tanh(\frac{\pi}{2}\sinh(t))\\ \hspace{30px}{\large\int_{\small -1}^{\small 1}}f(x)dx={\large\int_{\small -\infty}^{\small \infty}}f(x(t))x'(t)dt\\ \hspace{140px}x(t)= \tanh(\frac{\pi}{2}\sinh(t))\\ \hspace{140px}x'(t)={\large\frac{\frac{\pi}{2}\cosh(t)}{\cosh^2(\frac{\pi}{2}\sinh(t))}}\\ (3)\ Trapezoid\\ \hspace{20px}S\simeq{\large\int_{-t_a}^{t_a}}f(x(t))x'(t)dt=h{\large\displaystyle \sum_{{\small j=-}\frac{N}{2}}^{\frac{N}{2}}}f(x(jh))x'(jh)\\ \hspace{240px}h={\large\frac{2t_a}{N}}\\ \) |
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