Int - Maple Help

convert/Int

convert to inert integral representation

convert/int

convert to integral representation

 Calling Sequence convert( expr, Int, opts ); convert( expr, int, opts );

Parameters

 expr - valid Maple expression, equation, set, list, or similar structure opts - (optional) arguments as described in convert/to_special_function

Description

 • The convert(expr, Int) command replaces occurrences of mathematical functions by their integral representations using the inert form Int. It does not attempt to compute the integrals. This conversion facility accepts the optional arguments explained in convert/to_special_function so that you can restrict its application. By default, when possible, all mathematical functions are converted.
 • The convert(expr, int) command replaces occurrences of the inert form Int by the active form int, and then attempts to compute the integrals. This function also replaces any integral transforms,
 The 11 functions in the "integral_transforms" class are:
 $\left[{\mathrm{fourier}}{,}{\mathrm{fouriercos}}{,}{\mathrm{fouriersin}}{,}{\mathrm{hankel}}{,}{\mathrm{hilbert}}{,}{\mathrm{invfourier}}{,}{\mathrm{invhilbert}}{,}{\mathrm{invlaplace}}{,}{\mathrm{invmellin}}{,}{\mathrm{laplace}}{,}{\mathrm{mellin}}\right]$ (1)
 in an expression by the mathematically equivalent active form using int, and then attempts to compute the integrals. The exception to this is the inverse Laplace transform, which would require a convolution integral. It cannot be represented in Maple. By default, only the integral transforms mentioned above and occurrences of the inert Int are converted into active integrals. To override this default use the optional arguments described in convert/to_special_function.

Examples

 > $\mathrm{with}\left(\mathrm{inttrans}\right):$
 > $\mathrm{ee}≔\mathrm{fourier}\left(f\left(t\right),t,s\right)+\mathrm{mellin}\left(h\left(x\right),x,y\right)$
 ${\mathrm{ee}}{≔}{\mathrm{fourier}}{}\left({f}{}\left({t}\right){,}{t}{,}{s}\right){+}{\mathrm{mellin}}{}\left({h}{}\left({x}\right){,}{x}{,}{y}\right)$ (2)
 > $\mathrm{convert}\left(\mathrm{ee},\mathrm{int}\right)$
 ${{\int }}_{{-}{\mathrm{\infty }}}^{{\mathrm{\infty }}}{f}{}\left({t}\right){}{{ⅇ}}^{{-I}{}{t}{}{s}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}{+}{{\int }}_{{0}}^{{\mathrm{\infty }}}{h}{}\left({x}\right){}{{x}}^{{y}{-}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (3)
 > $\mathrm{convert}\left(\mathrm{ee},\mathrm{Int}\right)$
 ${{\int }}_{{-}{\mathrm{\infty }}}^{{\mathrm{\infty }}}{f}{}\left({t}\right){}{{ⅇ}}^{{-I}{}{t}{}{s}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}{+}{{\int }}_{{0}}^{{\mathrm{\infty }}}{h}{}\left({x}\right){}{{x}}^{{y}{-}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (4)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{Ei_related}\right)\left(z\right)$
 The 7 functions in the "Ei_related" class are:
 $\left[{\mathrm{Chi}}{}\left({z}\right){,}{\mathrm{Ci}}{}\left({z}\right){,}{\mathrm{Ei}}{}\left({z}\right){,}{\mathrm{Li}}{}\left({z}\right){,}{\mathrm{Shi}}{}\left({z}\right){,}{\mathrm{Si}}{}\left({z}\right){,}{\mathrm{Ssi}}{}\left({z}\right)\right]$ (5)
 > $\mathrm{map}\left(u↦u=\mathrm{convert}\left(u,\mathrm{Int}\right),\right)$
 $\left[{\mathrm{Chi}}{}\left({z}\right){=}{\mathrm{\gamma }}{+}{\mathrm{ln}}{}\left({z}\right){+}{{\int }}_{{0}}^{{z}}\frac{{\mathrm{cosh}}{}\left({\mathrm{_k1}}\right){-}{1}}{{\mathrm{_k1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}{,}{\mathrm{Ci}}{}\left({z}\right){=}{\mathrm{\gamma }}{+}{\mathrm{ln}}{}\left({z}\right){+}{{\int }}_{{0}}^{{z}}\frac{{\mathrm{cos}}{}\left({\mathrm{_k1}}\right){-}{1}}{{\mathrm{_k1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}{,}{\mathrm{Ei}}{}\left({z}\right){=}{\mathrm{Ei}}{}\left({z}\right){,}{\mathrm{Li}}{}\left({z}\right){=}{\mathrm{Li}}{}\left({z}\right){,}{\mathrm{Shi}}{}\left({z}\right){=}{{\int }}_{{0}}^{{z}}\frac{{\mathrm{sinh}}{}\left({\mathrm{_k1}}\right)}{{\mathrm{_k1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}{,}{\mathrm{Si}}{}\left({z}\right){=}{{\int }}_{{0}}^{{z}}\frac{{\mathrm{sin}}{}\left({\mathrm{_k1}}\right)}{{\mathrm{_k1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}{,}{\mathrm{Ssi}}{}\left({z}\right){=}{{\int }}_{{0}}^{{z}}\frac{{\mathrm{sin}}{}\left({\mathrm{_k1}}\right)}{{\mathrm{_k1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}{-}\frac{{\mathrm{\pi }}}{{2}}\right]$ (6)
 > $\mathrm{convert}\left(\left[-1\right],\mathrm{int}\right)$
 ${\mathrm{Ssi}}{}\left({z}\right){=}{\mathrm{Si}}{}\left({z}\right){-}\frac{{\mathrm{\pi }}}{{2}}$ (7)
 > $\mathrm{simplify}\left(\left(\mathrm{lhs}-\mathrm{rhs}\right)\left(\right)\right)$
 ${0}$ (8)

Note that some conversions are possible only under assumptions.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{Int},\mathrm{Β}\right)$
 * Partial match of "Int" against topic "integral_form".
 $\left[{\mathrm{Β}}{}\left({x}{,}{y}\right){=}{{\int }}_{{0}}^{{1}}{{\mathrm{_k1}}}^{{x}{-}{1}}{}{\left({1}{-}{\mathrm{_k1}}\right)}^{{y}{-}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}{,}{0}{<}{\mathrm{\Re }}{}\left({x}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left({y}\right)\right]$ (9)

Therefore, the following is not converted.

 > $\mathrm{convert}\left(\mathrm{Β}\left(a,z\right),\mathrm{Int}\right)$
 ${\mathrm{Β}}{}\left({a}{,}{z}\right)$ (10)

The conversion occurs only when you specify the proper assumptions.

 > $\mathrm{convert}\left(\mathrm{Β}\left(a,z\right),\mathrm{Int}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{And}\left(0<\mathrm{\Re }\left(a\right),0<\mathrm{\Re }\left(z\right)\right)$
 ${{\int }}_{{0}}^{{1}}{{\mathrm{_k1}}}^{{a}{-}{1}}{}{\left({1}{-}{\mathrm{_k1}}\right)}^{{z}{-}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}$ (11)