display information about a mathematical function organized in sections

return a table of information about a mathematical function

Parameters

 display - (optional) literal name;'display', since Maple 2016 it can be ommited, math_function - name of known mathematical function; see type/mathfunc table - (optional) literal name;'table', since Maple 2016 it returns a table of information about math_function

Description

 • The FunctionAdvisor(math_function) command returns the information regarding that function available to the system. The information that displays is organized in sections as shown in the examples.
 • Although the mathematical information in the sections is all computable (for instance, you can right click a formula and explore it, or just copy and paste it to work with it), it is sometimes convenient to have this information directly presented in a form that is more suitable for further computations. For this purpose use the optional argument table, in which case the same information is presented now as a table where the indices are the topics and the entries are the corresponding information.

Examples

The information about a mathematical function organized in closed sections

 > $\mathrm{FunctionAdvisor}\left(\mathrm{sin}\right)$

sin

describe

 ${\mathrm{sin}}{=}{\mathrm{sine function}}$

definition

 ${\mathrm{sin}}{}\left({z}\right){=}{-}\frac{{1}}{{2}}{}{I}{}\left({{ⅇ}}^{{I}{}{z}}{-}\frac{{1}}{{{ⅇ}}^{{I}{}{z}}}\right)$ ${\mathrm{with no restrictions on}}{}\left({z}\right)$

classify function

 ${\mathrm{trig}}$ ${\mathrm{elementary}}$

symmetries

 ${\mathrm{sin}}{}\left({-}{z}\right){=}{-}{\mathrm{sin}}{}\left({z}\right)$ ${\mathrm{sin}}{}\left(\stackrel{{&conjugate0;}}{{z}}\right){=}\stackrel{{&conjugate0;}}{{\mathrm{sin}}{}\left({z}\right)}$

periodicity

 ${\mathrm{sin}}{}\left({2}{}{\mathrm{\pi }}{}{m}{+}{z}\right){=}{\mathrm{sin}}{}\left({z}\right)$ $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{m}{::}{\mathrm{integer}}$

 ${\mathrm{sin}}{}\left({\mathrm{\pi }}{}{m}{+}{z}\right){=}{\left({-}{1}\right)}^{{m}}{}{\mathrm{sin}}{}\left({z}\right)$ $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{m}{::}{\mathrm{integer}}$

plot

singularities

 ${\mathrm{sin}}{}\left({z}\right)$ ${z}{=}{\mathrm{∞}}{+}{\mathrm{∞}}{}{I}$

branch points

 ${\mathrm{sin}}{}\left({z}\right)$ No branch points

branch cuts

 ${\mathrm{sin}}{}\left({z}\right)$ No branch cuts

special values

 ${\mathrm{sin}}{}\left(\frac{{1}}{{6}}{}{\mathrm{\pi }}\right){=}\frac{{1}}{{2}}$

 ${\mathrm{sin}}{}\left(\frac{{1}}{{4}}{}{\mathrm{\pi }}\right){=}\frac{{1}}{{2}}{}\sqrt{{2}}$

 ${\mathrm{sin}}{}\left(\frac{{1}}{{3}}{}{\mathrm{\pi }}\right){=}\frac{{1}}{{2}}{}\sqrt{{3}}$

 ${\mathrm{sin}}{}\left({\mathrm{∞}}\right){=}{\mathrm{undefined}}$

 ${\mathrm{sin}}{}\left({\mathrm{∞}}{}{I}\right){=}{\mathrm{∞}}{}{I}$

 ${\mathrm{sin}}{}\left({\mathrm{\pi }}{}{n}\right){=}{0}$ $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{n}{::}{\mathrm{integer}}$

 ${\mathrm{sin}}{}\left(\frac{{1}}{{2}}{}\left({2}{}{n}{+}{1}\right){}{\mathrm{\pi }}\right){=}{-}{1}$ $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{n}{::}{\mathrm{odd}}$

 ${\mathrm{sin}}{}\left(\frac{{1}}{{2}}{}\left({2}{}{n}{+}{1}\right){}{\mathrm{\pi }}\right){=}{1}$ $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{n}{::}{\mathrm{even}}$

identities

 ${\mathrm{sin}}{}\left({\mathrm{arcsin}}{}\left({z}\right)\right){=}{z}$ ${\mathrm{sin}}{}\left({z}\right){=}{-}{\mathrm{sin}}{}\left({-}{z}\right)$ ${\mathrm{sin}}{}\left({z}\right){=}{2}{}{\mathrm{sin}}{}\left(\frac{{1}}{{2}}{}{z}\right){}{\mathrm{cos}}{}\left(\frac{{1}}{{2}}{}{z}\right)$ ${\mathrm{sin}}{}\left({z}\right){=}\frac{{1}}{{\mathrm{csc}}{}\left({z}\right)}$ ${\mathrm{sin}}{}\left({z}\right){=}\frac{{2}{}{\mathrm{tan}}{}\left(\frac{{1}}{{2}}{}{z}\right)}{{1}{+}{{\mathrm{tan}}{}\left(\frac{{1}}{{2}}{}{z}\right)}^{{2}}}$ ${\mathrm{sin}}{}\left({z}\right){=}{-}\frac{{1}}{{2}}{}{I}{}\left({{ⅇ}}^{{I}{}{z}}{-}{{ⅇ}}^{{-}{I}{}{z}}\right)$ ${{\mathrm{sin}}{}\left({z}\right)}^{{2}}{=}{1}{-}{{\mathrm{cos}}{}\left({z}\right)}^{{2}}$ ${{\mathrm{sin}}{}\left({z}\right)}^{{2}}{=}\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{\mathrm{cos}}{}\left({2}{}{z}\right)$

sum form

 ${\mathrm{sin}}{}\left({z}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{\mathrm{_k1}}}{}{{z}}^{{2}{}{\mathrm{_k1}}{+}{1}}}{\left({2}{}{\mathrm{_k1}}{+}{1}\right){!}}$ ${\mathrm{with no restrictions on}}{}\left({z}\right)$

series

 ${\mathrm{series}}{}\left({\mathrm{sin}}{}\left({z}\right){,}{z}{,}{4}\right){=}{z}{-}\frac{{1}}{{6}}{}{{z}}^{{3}}{+}{\mathrm{O}}\left({{z}}^{{5}}\right)$

integral form

 ${\mathrm{sin}}{}\left({z}\right){=}\frac{{z}{}\left({{∫}}_{{0}}^{{1}}{{ⅇ}}^{{2}{}{I}{}{\mathrm{_t1}}{}{z}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_t1}}\right)}{{{ⅇ}}^{{I}{}{z}}}$ ${\mathrm{with no restrictions on}}{}\left({z}\right)$

differentiation rule

 $\frac{{ⅆ}}{{ⅆ}{z}}{}{\mathrm{sin}}{}\left({z}\right){=}{\mathrm{cos}}{}\left({z}\right)$

 $\frac{{{ⅆ}}^{{n}}}{{ⅆ}{{z}}^{{n}}}{}{\mathrm{sin}}{}\left({z}\right){=}{\mathrm{sin}}{}\left({z}{+}\frac{{1}}{{2}}{}{n}{}{\mathrm{\pi }}\right)$

DE

${f}{}\left({z}\right){=}{\mathrm{sin}}{}\left({z}\right)$

 $\frac{{ⅆ}}{{ⅆ}{z}}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}{}{f}{}\left({z}\right)\right){=}{-}{f}{}\left({z}\right)$

To get a Maple table structure with this same information use the table keyword (to avoid verbosity, use the option quiet)

 > $\mathrm{sin_info}≔\mathrm{FunctionAdvisor}\left(\mathrm{table},\mathrm{sin},\mathrm{quiet}\right)$
 ${\mathrm{sin_info}}{≔}{\mathrm{table}}\left(\left[{"asymptotic_expansion"}{=}\left(\right){,}{"DE"}{=}\left[{f}\left({z}\right){=}{\mathrm{sin}}\left({z}\right){,}\left[\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}{}{f}\left({z}\right){=}{-}{f}\left({z}\right)\right]\right]{,}{"definition"}{=}\left[{\mathrm{sin}}\left({z}\right){=}{-}\frac{{1}}{{2}}{}{I}{}\left({{ⅇ}}^{{I}{}{z}}{-}\frac{{1}}{{{ⅇ}}^{{I}{}{z}}}\right){,}{\mathrm{with no restrictions on}}\left({z}\right)\right]{,}{"sum_form"}{=}\left[{\mathrm{sin}}\left({z}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{\mathrm{_k1}}}{}{{z}}^{{2}{}{\mathrm{_k1}}{+}{1}}}{\left({2}{}{\mathrm{_k1}}{+}{1}\right){!}}{,}{\mathrm{with no restrictions on}}\left({z}\right)\right]{,}{"describe"}{=}\left({\mathrm{sin}}{=}{\mathrm{sine function}}\right){,}{"calling_sequence"}{=}{\mathrm{sin}}\left({z}\right){,}{"classify_function"}{=}\left({\mathrm{trig}}{,}{\mathrm{elementary}}\right){,}{"symmetries"}{=}\left[{\mathrm{sin}}\left({-}{z}\right){=}{-}{\mathrm{sin}}\left({z}\right){,}{\mathrm{sin}}\left(\stackrel{{&conjugate0;}}{{z}}\right){=}\stackrel{{&conjugate0;}}{{\mathrm{sin}}\left({z}\right)}\right]{,}{"branch_points"}{=}\left[{\mathrm{sin}}\left({z}\right){,}{"No branch points"}\right]{,}{"integral_form"}{=}\left[{\mathrm{sin}}\left({z}\right){=}\frac{{z}{}\left({{∫}}_{{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]{,}{"differentiation_rule"}{=}\left(\frac{{ⅆ}}{{ⅆ}{z}}{}{\mathrm{sin}}\left({z}\right){=}{\mathrm{cos}}\left({z}\right){,}\frac{{{ⅆ}}^{{n}}}{{ⅆ}{{z}}^{{n}}}{}{\mathrm{sin}}\left({z}\right){=}{\mathrm{sin}}\left({z}{+}\frac{{1}}{{2}}{}{n}{}{\mathrm{π}}\right)\right){,}{"singularities"}{=}\left[{\mathrm{sin}}\left({z}\right){,}{z}{=}{\mathrm{∞}}{+}{\mathrm{∞}}{}{I}\right]{,}{"branch_cuts"}{=}\left[{\mathrm{sin}}\left({z}\right){,}{"No branch cuts"}\right]{,}{"periodicity"}{=}\left[\left[{\mathrm{sin}}\left({2}{}{\mathrm{π}}{}{m}{+}{z}\right){=}{\mathrm{sin}}\left({z}\right){,}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{m}{::}{\mathrm{integer}}\right]{,}\left[{\mathrm{sin}}\left({\mathrm{π}}{}{m}{+}{z}\right){=}{\left({-}{1}\right)}^{{m}}{}{\mathrm{sin}}\left({z}\right){,}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{m}{::}{\mathrm{integer}}\right]\right]{,}{"series"}{=}\left({\mathrm{series}}\left({\mathrm{sin}}\left({z}\right){,}{z}{,}{4}\right){=}{z}{-}\frac{{1}}{{6}}{}{{z}}^{{3}}{+}{\mathrm{O}}\left({{z}}^{{5}}\right)\right){,}{"identities"}{=}\left[{\mathrm{sin}}\left({\mathrm{arcsin}}\left({z}\right)\right){=}{z}{,}{\mathrm{sin}}\left({z}\right){=}{-}{\mathrm{sin}}\left({-}{z}\right){,}{\mathrm{sin}}\left({z}\right){=}{2}{}{\mathrm{sin}}\left(\frac{{1}}{{2}}{}{z}\right){}{\mathrm{cos}}\left(\frac{{1}}{{2}}{}{z}\right){,}{\mathrm{sin}}\left({z}\right){=}\frac{{1}}{{\mathrm{csc}}\left({z}\right)}{,}{\mathrm{sin}}\left({z}\right){=}\frac{{2}{}{\mathrm{tan}}\left(\frac{{1}}{{2}}{}{z}\right)}{{1}{+}{{\mathrm{tan}}\left(\frac{{1}}{{2}}{}{z}\right)}^{{2}}}{,}{\mathrm{sin}}\left({z}\right){=}{-}\frac{{1}}{{2}}{}{I}{}\left({{ⅇ}}^{{I}{}{z}}{-}{{ⅇ}}^{{-}{I}{}{z}}\right){,}{{\mathrm{sin}}\left({z}\right)}^{{2}}{=}{1}{-}{{\mathrm{cos}}\left({z}\right)}^{{2}}{,}{{\mathrm{sin}}\left({z}\right)}^{{2}}{=}\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{\mathrm{cos}}\left({2}{}{z}\right)\right]{,}{"special_values"}{=}\left[{\mathrm{sin}}\left(\frac{{1}}{{6}}{}{\mathrm{π}}\right){=}\frac{{1}}{{2}}{,}{\mathrm{sin}}\left(\frac{{1}}{{4}}{}{\mathrm{π}}\right){=}\frac{{1}}{{2}}{}\sqrt{{2}}{,}{\mathrm{sin}}\left(\frac{{1}}{{3}}{}{\mathrm{π}}\right){=}\frac{{1}}{{2}}{}\sqrt{{3}}{,}{\mathrm{sin}}\left({\mathrm{∞}}\right){=}{\mathrm{undefined}}{,}{\mathrm{sin}}\left({\mathrm{∞}}{}{I}\right){=}{\mathrm{∞}}{}{I}{,}\left[{\mathrm{sin}}\left({\mathrm{π}}{}{n}\right){=}{0}{,}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{n}{::}{\mathrm{integer}}\right]{,}\left[{\mathrm{sin}}\left(\frac{{1}}{{2}}{}\left({2}{}{n}{+}{1}\right){}{\mathrm{π}}\right){=}{-}{1}{,}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{n}{::}{\mathrm{odd}}\right]{,}\left[{\mathrm{sin}}\left(\frac{{1}}{{2}}{}\left({2}{}{n}{+}{1}\right){}{\mathrm{π}}\right){=}{1}{,}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{And}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{n}{::}{\mathrm{even}}\right]\right]\right]\right)$ (1)

You can now access the information indexing with the FunctionAdvisor topics

 > ${\mathrm{sin_info}}_{"differentiation_rule"}$
 $\frac{{ⅆ}}{{ⅆ}{z}}{}{\mathrm{sin}}\left({z}\right){=}{\mathrm{cos}}\left({z}\right){,}\frac{{{ⅆ}}^{{n}}}{{ⅆ}{{z}}^{{n}}}{}{\mathrm{sin}}\left({z}\right){=}{\mathrm{sin}}\left({z}{+}\frac{{1}}{{2}}{}{n}{}{\mathrm{π}}\right)$ (2)
 > 

Compatibility

 • The FunctionAdvisor/table command was introduced in Maple 2016.