Overview - Maple Help

provide information on mathematical functions in general

Parameters

 topics - literal name; 'topics'; specify that the FunctionAdvisor command return the topics for which information is available quiet - (optional) literal name; 'quiet'; specify that only the computational result in Maple syntax is returned Topic - (optional) name; FunctionAdvisor topic function - name; mathematical function or function class. For some topics, you can specify multiple mathematical functions opts - (optional) topic-specific options

Description

 • The FunctionAdvisor() command returns basic instructions for the use of the FunctionAdvisor function.
 • The FunctionAdvisor(function) command returns a summary of information related to the function function.
 • The FunctionAdvisor(Topic, function) command returns information related to the topic Topic for the function function.
 • The requirement concerning mathematical functions is not just computational. Typically, you need supporting information on definitions, identities, possible simplifications, integral forms, different types of series expansions, and mathematical properties in general. This information is in handbooks of mathematical functions like the one by Abramowitz and Stegun. You can now access this information directly from Maple, using the routines in the MathematicalFunctions package and the FunctionAdvisor command. This command is particularly useful when studying, teaching, and solving problems where mathematical function properties are relevant.
 • Using the FunctionAdvisor command, you can access mathematical language information easily that is both readable and directly usable in Maple mathematical computations. The FunctionAdvisor command provides information on the following topics.

 The FunctionAdvisor command provides information on the following mathematical functions.

 Like the conversion facility for mathematical functions, the FunctionAdvisor command also works with the concept of function classes and considers assumptions on the function parameters, if any. The FunctionAdvisor command provides information on the following function classes.

 • The FunctionAdvisor command can be considered to be between a help and a computational special function facility. Due to the wide range of information this command can handle and in order to facilitate its use, it includes two distinctive features:
 – If you call the FunctionAdvisor command without arguments, it returns information that you can follow until the appropriate information displays.
 – If you call the FunctionAdvisor command with a topic or function misspelled, but a match exists, it returns the information with a warning message.
 So, you do not have to remember the exact Maple name of each mathematical function or the FunctionAdvisor topic. To avoid the warning messages displayed when the topic of function is misspelled, and the FunctionAdvisor verbosity in general, specify the optional argument quiet.

Examples

The following example uses the FunctionAdvisor command with no arguments specified.

 > $\mathrm{FunctionAdvisor}\left(\right)$
 The usage is as follows:     > FunctionAdvisor( topic, function, ... ); where 'topic' indicates the subject on which advice is required, 'function' is the name of a Maple function, and '...' represents possible additional input depending on the 'topic' chosen. To list the possible topics:     > FunctionAdvisor( topics ); A short form usage,     > FunctionAdvisor( function ); with just the name of the function is also available and displays a summary of information about the function.
 > $\mathrm{FunctionAdvisor}\left(\mathrm{topics}\right)$
 The topics on which information is available are:
 $\left[{\mathrm{DE}}{,}{\mathrm{analytic_extension}}{,}{\mathrm{asymptotic_expansion}}{,}{\mathrm{branch_cuts}}{,}{\mathrm{branch_points}}{,}{\mathrm{calling_sequence}}{,}{\mathrm{class_members}}{,}{\mathrm{classify_function}}{,}{\mathrm{definition}}{,}{\mathrm{describe}}{,}{\mathrm{differentiation_rule}}{,}{\mathrm{function_classes}}{,}{\mathrm{identities}}{,}{\mathrm{integral_form}}{,}{\mathrm{known_functions}}{,}{\mathrm{periodicity}}{,}{\mathrm{plot}}{,}{\mathrm{relate}}{,}{\mathrm{required_assumptions}}{,}{\mathrm{series}}{,}{\mathrm{singularities}}{,}{\mathrm{special_values}}{,}{\mathrm{specialize}}{,}{\mathrm{sum_form}}{,}{\mathrm{symmetries}}{,}{\mathrm{synonyms}}{,}{\mathrm{table}}\right]$ (1)

To avoid all FunctionAdvisor verbosity, specify the optional argument quiet.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{function_classes},\mathrm{quiet}\right)$
 $\left[{\mathrm{trig}}{,}{\mathrm{trigh}}{,}{\mathrm{arctrig}}{,}{\mathrm{arctrigh}}{,}{\mathrm{elementary}}{,}{\mathrm{GAMMA_related}}{,}{\mathrm{Psi_related}}{,}{\mathrm{Kelvin}}{,}{\mathrm{Airy}}{,}{\mathrm{Hankel}}{,}{\mathrm{Bessel_related}}{,}{\mathrm{0F1}}{,}{\mathrm{orthogonal_polynomials}}{,}{\mathrm{Ei_related}}{,}{\mathrm{erf_related}}{,}{\mathrm{Kummer}}{,}{\mathrm{Whittaker}}{,}{\mathrm{Cylinder}}{,}{\mathrm{1F1}}{,}{\mathrm{Elliptic_related}}{,}{\mathrm{Legendre}}{,}{\mathrm{Chebyshev}}{,}{\mathrm{2F1}}{,}{\mathrm{Lommel}}{,}{\mathrm{Struve_related}}{,}{\mathrm{hypergeometric}}{,}{\mathrm{Jacobi_related}}{,}{\mathrm{InverseJacobi_related}}{,}{\mathrm{Elliptic_doubly_periodic}}{,}{\mathrm{Weierstrass_related}}{,}{\mathrm{Zeta_related}}{,}{\mathrm{complex_components}}{,}{\mathrm{piecewise_related}}{,}{\mathrm{Other}}{,}{\mathrm{Bell}}{,}{\mathrm{Heun}}{,}{\mathrm{Appell}}{,}{\mathrm{trigall}}{,}{\mathrm{arctrigall}}{,}{\mathrm{Polylog_related}}{,}{\mathrm{integral_transforms}}\right]$ (2)

The FunctionAdvisor command can return information ranging from general information, for example, "the Maple names for the Bessel functions",

 > $\mathrm{FunctionAdvisor}\left(\mathrm{bess}\right)$
 * Partial match of "bess" against topic "Bessel_related". The 14 functions in the "Bessel_related" class are:
 $\left[{\mathrm{AiryAi}}{,}{\mathrm{AiryBi}}{,}{\mathrm{BesselI}}{,}{\mathrm{BesselJ}}{,}{\mathrm{BesselK}}{,}{\mathrm{BesselY}}{,}{\mathrm{HankelH1}}{,}{\mathrm{HankelH2}}{,}{\mathrm{KelvinBei}}{,}{\mathrm{KelvinBer}}{,}{\mathrm{KelvinHei}}{,}{\mathrm{KelvinHer}}{,}{\mathrm{KelvinKei}}{,}{\mathrm{KelvinKer}}\right]$ (3)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{describe},\mathrm{BesselK}\right)$
 ${\mathrm{BesselK}}{=}{\mathrm{Modified Bessel function of the second kind}}$ (4)

to more complicated relationships between mathematical functions and their identities, computed using the Maple internal knowledge database and related algorithms.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{tan}\right)$
 $\left[{\mathrm{tan}}{}\left({z}\right){=}{\sum }_{{\mathrm{_k1}}{=}{1}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{bernoulli}}{}\left({2}{}{\mathrm{_k1}}\right){}{\left({-1}\right)}^{{\mathrm{_k1}}}{}{{z}}^{{-}{1}{+}{2}{}{\mathrm{_k1}}}{}\left({{4}}^{{\mathrm{_k1}}}{-}{{16}}^{{\mathrm{_k1}}}\right)}{{\mathrm{\Gamma }}{}\left({2}{}{\mathrm{_k1}}{+}{1}\right)}{,}\left|{z}\right|{<}\frac{{\mathrm{\pi }}}{{2}}\right]$ (5)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{integral_form},\mathrm{Β}\right)$
 $\left[{\mathrm{Β}}{}\left({x}{,}{y}\right){=}{{\int }}_{{0}}^{{1}}{{\mathrm{_k1}}}^{{x}{-}{1}}{}{\left({1}{-}{\mathrm{_k1}}\right)}^{{y}{-}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_k1}}{,}{0}{<}{\mathrm{\Re }}{}\left({x}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left({y}\right)\right]$ (6)
 > $\mathrm{sp_eq}≔\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{HermiteH},\mathrm{KummerU}\right)$
 ${\mathrm{sp_eq}}{≔}\left[{\mathrm{HermiteH}}{}\left({a}{,}{z}\right){=}{{2}}^{{a}}{}{\mathrm{KummerU}}{}\left({-}\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{,}{{z}}^{{2}}\right){,}{0}{<}{\mathrm{\Re }}{}\left({z}\right){\vee }\left({\mathrm{\Re }}{}\left({z}\right){=}{0}{\wedge }{0}{<}{\mathrm{\Im }}{}\left({z}\right)\right)\right]$ (7)

If you only specify function names, the parameters entering the mathematical formulas are all local variables. For example, the previous formula uses local instances of a and z and therefore

 > $\mathrm{has}\left(\left[\mathrm{sp_eq}\right],\left\{a,z\right\}\right)$
 ${\mathrm{false}}$ (8)

You can override this behavior by passing the function with the parameters. For example, you can first retrieve the calling sequence then pass $\mathrm{EllipticF}\left(z,k\right)$.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{syntax},\mathrm{EllipticF}\right)$
 ${\mathrm{EllipticF}}{}\left({z}{,}{k}\right)$ (9)
 > $\mathrm{EF_and_DE}≔\mathrm{FunctionAdvisor}\left(\mathrm{DE},\mathrm{EllipticF}\left(z,k\right)\right)$
 ${\mathrm{EF_and_DE}}{≔}\left[{f}{}\left({z}{,}{k}\right){=}{\mathrm{EllipticF}}{}\left({z}{,}{k}\right){,}\left[\frac{{{\partial }}^{{2}}}{{\partial }{{k}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}{,}{k}\right){=}\frac{\left({-}{1}{-}{3}{}{{k}}^{{4}}{}{{z}}^{{2}}{+}\left({{z}}^{{2}}{+}{3}\right){}{{k}}^{{2}}\right){}\left(\frac{{\partial }}{{\partial }{k}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}{,}{k}\right)\right)}{{{k}}^{{5}}{}{{z}}^{{2}}{+}\left({-}{{z}}^{{2}}{-}{1}\right){}{{k}}^{{3}}{+}{k}}{+}\frac{\left({{z}}^{{3}}{-}{z}\right){}\left(\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}{,}{k}\right)\right)}{\left({{k}}^{{4}}{-}{{k}}^{{2}}\right){}{{z}}^{{2}}{-}{{k}}^{{2}}{+}{1}}{+}\frac{\left({-}{{k}}^{{2}}{}{{z}}^{{2}}{+}{1}\right){}{f}{}\left({z}{,}{k}\right)}{{{k}}^{{4}}{}{{z}}^{{2}}{-}{{k}}^{{2}}{}{{z}}^{{2}}{-}{{k}}^{{2}}{+}{1}}{,}\frac{{{\partial }}^{{2}}}{{\partial }{k}{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}{,}{k}\right){=}{-}\frac{\left(\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}{,}{k}\right)\right){}{k}{}{{z}}^{{2}}}{{{k}}^{{2}}{}{{z}}^{{2}}{-}{1}}{,}\frac{{{\partial }}^{{2}}}{{\partial }{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}{,}{k}\right){=}\frac{\left({-}{2}{}{{k}}^{{2}}{}{{z}}^{{3}}{+}\left({{k}}^{{2}}{+}{1}\right){}{z}\right){}\left(\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}{,}{k}\right)\right)}{{1}{+}{{k}}^{{2}}{}{{z}}^{{4}}{+}\left({-}{{k}}^{{2}}{-}{1}\right){}{{z}}^{{2}}}\right]\right]$ (10)
 > $\mathrm{map2}\left(\mathrm{has},\left[\mathrm{EF_and_DE}\right],\left[z,k\right]\right)$
 $\left[{\mathrm{true}}{,}{\mathrm{true}}\right]$ (11)

The information returned by the FunctionAdvisor command can be used for further computations. For example, you can verify that the first operand EF_and_DE, that is, EllipticF, is a solution of the second operand, a PDE system, or further represent the function in differently.

 > $\mathrm{pdetest}\left(\mathrm{EF_and_DE}\left[1\right],\mathrm{EF_and_DE}\left[2\right]\right)$
 $\left[{0}{,}{0}{,}{0}\right]$ (12)
 > $\mathrm{EF_and_DE}\left[1\right]$
 ${f}{}\left({z}{,}{k}\right){=}{\mathrm{EllipticF}}{}\left({z}{,}{k}\right)$ (13)
 > $\mathrm{convert}\left(\mathrm{EF_and_DE}\left[1\right],\mathrm{Int}\right)$
 ${f}{}\left({z}{,}{k}\right){=}{{\int }}_{{0}}^{{z}}\frac{{1}}{\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{+}{1}}{}\sqrt{{-}{{k}}^{{2}}{}{{\mathrm{_α1}}}^{{2}}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_α1}}$ (14)

Use the FunctionAdvisor command to return a presentation with sections of information for the arccot function.

 > $\mathrm{FunctionAdvisor}\left(\mathrm{arccot},\mathrm{quiet}\right):$

arccot

describe

 ${\mathrm{arccot}}{=}{\mathrm{inverse cotangent function}}$

definition

 ${\mathrm{arccot}}{}\left({z}\right){=}\frac{{\mathrm{\pi }}}{{2}}{-}\frac{{I}{}\left({\mathrm{ln}}{}\left({1}{-}{I}{}{z}\right){-}{\mathrm{ln}}{}\left({1}{+}{I}{}{z}\right)\right)}{{2}}$ ${\mathrm{with no restrictions on}}{}\left({z}\right)$

classify function

 ${\mathrm{arctrig}}$ ${\mathrm{elementary}}$

symmetries

 ${\mathrm{arccot}}{}\left({-}{z}\right){=}{\mathrm{\pi }}{-}{\mathrm{arccot}}{}\left({z}\right)$

 ${\mathrm{arccot}}{}\left(\stackrel{{&conjugate0;}}{{z}}\right){=}\stackrel{{&conjugate0;}}{{\mathrm{arccot}}{}\left({z}\right)}$ ${\mathbf{not}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left({z}{\in }{\mathrm{ComplexRange}}{}\left({-}{\mathrm{\infty }}{}{I}{,}{-I}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{z}{\in }{\mathrm{ComplexRange}}{}\left({I}{,}{\mathrm{\infty }}{}{I}\right)\right)$

periodicity

 ${\mathrm{arccot}}{}\left({z}\right)$ No periodicity

plot

singularities

 ${\mathrm{arccot}}{}\left({z}\right)$ ${z}{=}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}$

branch points

 ${\mathrm{arccot}}{}\left({z}\right)$ ${z}{\in }\left[{-I}{,}{I}\right]$

branch cuts

 ${\mathrm{arccot}}{}\left({z}\right)$ ${z}{\in }{\mathrm{ComplexRange}}{}\left({-}{\mathrm{\infty }}{}{I}{,}{-I}\right){\vee }{z}{\in }{\mathrm{ComplexRange}}{}\left({I}{,}{\mathrm{\infty }}{}{I}\right)$

special values

 ${\mathrm{arccot}}{}\left({-1}\right){=}\frac{{3}{}{\mathrm{\pi }}}{{4}}$ ${\mathrm{arccot}}{}\left({-}\frac{\sqrt{{3}}}{{3}}\right){=}\frac{{2}{}{\mathrm{\pi }}}{{3}}$ ${\mathrm{arccot}}{}\left({-}\sqrt{{3}}\right){=}\frac{{5}{}{\mathrm{\pi }}}{{6}}$ ${\mathrm{arccot}}{}\left({0}\right){=}\frac{{\mathrm{\pi }}}{{2}}$ ${\mathrm{arccot}}{}\left(\sqrt{{3}}\right){=}\frac{{\mathrm{\pi }}}{{6}}$ ${\mathrm{arccot}}{}\left(\frac{\sqrt{{3}}}{{3}}\right){=}\frac{{\mathrm{\pi }}}{{3}}$ ${\mathrm{arccot}}{}\left({1}\right){=}\frac{{\mathrm{\pi }}}{{4}}$ ${\mathrm{arccot}}{}\left({\mathrm{\infty }}\right){=}{0}$ ${\mathrm{arccot}}{}\left({-}{\mathrm{\infty }}\right){=}{\mathrm{\pi }}$

identities

 ${\mathrm{cot}}{}\left({\mathrm{arccot}}{}\left({z}\right)\right){=}{z}$ ${\mathrm{cot}}{}\left({\mathrm{arccot}}{}\left({z}\right){+}{\mathrm{arccot}}{}\left(\right)\right)$