FunctionAdvisor/integral_form - Maple Programming Help

return the integral form of a given mathematical function

 Calling Sequence FunctionAdvisor(integral_form, math_function)

Parameters

 integral_form - literal name; 'integral_form' math_function - Maple name of mathematical function

Description

 • The FunctionAdvisor(integral_form, math_function) command returns the integral form for the function, if it exists.

Examples

 > $\mathrm{FunctionAdvisor}\left(\mathrm{integral_form},\mathrm{sin}\right)$
 $\left[{\mathrm{sin}}{}\left({z}\right){=}\frac{{z}{}\left({{\int }}_{{0}}^{{1}}{{ⅇ}}^{{2}{}{I}{}{\mathrm{_t1}}{}{z}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_t1}}\right)}{{{ⅇ}}^{{I}{}{z}}}{,}{\mathrm{with no restrictions on}}{}\left({z}\right)\right]$ (1)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{integral_form},\mathrm{Β}\left(a,z\right)\right)$
 $\left[{\mathrm{Β}}{}\left({a}{,}{z}\right){=}{{\int }}_{{0}}^{{1}}{{\mathrm{_k1}}}^{{a}{-}{1}}{}{\left({1}{-}{\mathrm{_k1}}\right)}^{{z}{-}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}{,}{0}{<}{\mathrm{\Re }}{}\left({a}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left({z}\right)\right]$ (2)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{describe},\mathrm{EllipticE}\right)$
 ${\mathrm{EllipticE}}{=}{\mathrm{incomplete or complete elliptic integral of the second kind}}$ (3)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{integral},\mathrm{EllipticE}\right)$
 * Partial match of "integral" against topic "integral_form".
 $\left[{\mathrm{EllipticE}}{}\left({k}\right){=}{{\int }}_{{0}}^{{1}}\frac{\sqrt{{-}{{k}}^{{2}}{}{{\mathrm{_α1}}}^{{2}}{+}{1}}}{\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_α1}}{,}{\mathrm{with no restrictions on}}{}\left({k}\right)\right]{,}\left[{\mathrm{EllipticE}}{}\left({z}{,}{k}\right){=}{{\int }}_{{0}}^{{z}}\frac{\sqrt{{-}{{k}}^{{2}}{}{{\mathrm{_α1}}}^{{2}}{+}{1}}}{\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_α1}}{,}{\mathrm{with no restrictions on}}{}\left({z}{,}{k}\right)\right]$ (4)
 > $\mathrm{ex1}≔\mathrm{FunctionAdvisor}\left(\mathrm{integral},\mathrm{BesselJ}\right)$
 * Partial match of "integral" against topic "integral_form".
 ${\mathrm{ex1}}{≔}\left[{\mathrm{BesselJ}}{}\left({a}{,}{z}\right){=}{{\int }}_{{-}{\mathrm{\pi }}}^{{\mathrm{\pi }}}\frac{{1}}{{2}{}{\mathrm{\pi }}{}{{ⅇ}}^{{I}{}{a}{}{\mathrm{_k1}}{-}{I}{}{z}{}{\mathrm{sin}}{}\left({\mathrm{_k1}}\right)}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}{,}{a}{::}{?}\right]{,}\left[{\mathrm{BesselJ}}{}\left({a}{,}{z}\right){=}{{\int }}_{{0}}^{{\mathrm{\infty }}}{-}\frac{{2}{}{\mathrm{sin}}{}\left({-}{z}{}{\mathrm{cosh}}{}\left({\mathrm{_k1}}\right){+}\frac{{\mathrm{\pi }}{}{a}}{{2}}\right){}{\mathrm{cosh}}{}\left({a}{}{\mathrm{_k1}}\right)}{{\mathrm{\pi }}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}{,}{z}{::}{\mathrm{real}}\right]{,}\left[{\mathrm{BesselJ}}{}\left({a}{,}{z}\right){=}{{\int }}_{{0}}^{{\mathrm{\pi }}}\frac{{\mathrm{cos}}{}\left({a}{}{\mathrm{_k1}}{-}{z}{}{\mathrm{sin}}{}\left({\mathrm{_k1}}\right)\right)}{{\mathrm{\pi }}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}{-}\frac{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{a}\right){}\left({{\int }}_{{0}}^{{\mathrm{\infty }}}\frac{{1}}{{{ⅇ}}^{{I}{}{\mathrm{_k1}}{+}{z}{}{\mathrm{sinh}}{}\left({\mathrm{_k1}}\right)}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}\right)}{{\mathrm{\pi }}}{,}{0}{<}{\mathrm{\Re }}{}\left({z}\right)\right]{,}\left[{\mathrm{BesselJ}}{}\left({a}{,}{z}\right){=}\frac{{{z}}^{{a}}{}\left({{\int }}_{{0}}^{{1}}{{ⅇ}}^{{2}{}{I}{}{\mathrm{_t1}}{}{z}}{}{{\mathrm{_t1}}}^{{-}\frac{{1}}{{2}}{+}{a}}{}{\left({1}{-}{\mathrm{_t1}}\right)}^{{-}\frac{{1}}{{2}}{+}{a}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_t1}}\right){}{{2}}^{{1}{+}{2}{}{a}}}{{2}{}{{2}}^{{a}}{}{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{a}\right){}{{ⅇ}}^{{I}{}{z}}{}\sqrt{{\mathrm{\pi }}}}{,}{0}{<}\frac{{1}}{{2}}{+}{\mathrm{\Re }}{}\left({a}\right)\right]$ (5)

The variables used by the FunctionAdvisor command to create the function calling sequences are local variables. Therefore, the previous example does not depend on a or z.

 > $\mathrm{depends}\left(\left[\mathrm{ex1}\right],a\right),\mathrm{depends}\left(\left[\mathrm{ex1}\right],z\right)$
 ${\mathrm{false}}{,}{\mathrm{false}}$ (6)

To make the FunctionAdvisor command return resulting using global variables, pass the function call itself.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{calling},\mathrm{EllipticF}\right)$
 * Partial match of "calling" against topic "calling_sequence".
 ${\mathrm{EllipticF}}{}\left({z}{,}{k}\right)$ (7)
 > $\mathrm{ex2}≔\mathrm{FunctionAdvisor}\left(\mathrm{integral},\mathrm{EllipticF}\left(a,z\right)\right)$
 * Partial match of "integral" against topic "integral_form".
 ${\mathrm{ex2}}{≔}\left[{\mathrm{EllipticF}}{}\left({a}{,}{z}\right){=}{{\int }}_{{0}}^{{a}}\frac{{1}}{\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{+}{1}}{}\sqrt{{-}{{z}}^{{2}}{}{{\mathrm{_α1}}}^{{2}}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_α1}}{,}{\mathrm{with no restrictions on}}{}\left({a}{,}{z}\right)\right]$ (8)
 > $\mathrm{depends}\left(\mathrm{ex2},a\right),\mathrm{depends}\left(\mathrm{ex2},z\right)$
 ${\mathrm{true}}{,}{\mathrm{true}}$ (9)