Double exponential integration (-∞,b) Calculator

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Calculates the integral of the given function f(x) over the semi-infinite interval (-∞,b) using the Double exponential formula.

This method is suitable for the function with algebraic decay at infinity.
The integrand f(x) is assumed to be analytic and non-periodic except for the endpoint.

\( \normalsize Double\ exponential\ integration\\
(1)\ x\rightarrow\ e^{{\normalsize\frac{\pi}{2}}\sinh(t)}\\
\ {\large\int_{\tiny -\infty}^{\tiny b}}f(x)dx={\large\int_{\tiny 0}^{\tiny\infty}}f(-(x-b))dx={\large\int_{\tiny -\infty}^{\tiny\infty}}f(-(x(t)-b))x'(t)dt\\
\hspace{140px}x(t)=e^{{\normalsize\frac{\pi}{2}}\sinh(t)}\\
\hspace{140px}x'(t)=e^{{\normalsize\frac{\pi}{2}}\sinh(t)}\frac{\pi}{2}\cosh(t)\\
(2)\ Trapezoid\\
\hspace{20px}S\simeq{\large\int_{-t_a}^{t_a}}f(-y(t))y'(t)dt=h{\large\displaystyle \sum_{{\small j=-}\frac{N}{2}}^{\frac{N}{2}}}f(-y(jh))y'(jh)\\
\hspace{240px}h={\large\frac{2t_a}{N}}\\
\)

Double exponential integration (-∞,b) Calculator

Calculates the integral of the given function f(x) over the semi-infinite interval (-∞,b) using the Double exponential formula.

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Double exponential integration (-∞,b) Calculator

Calculates the integral of the given function f(x) over the semi-infinite interval (-∞,b) using the Double exponential formula.

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History

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