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FunctionAdvisor - provide information on mathematical functions in general

Calling Sequence

FunctionAdvisor()

FunctionAdvisor(topics, quiet)

FunctionAdvisor(Topic, function, quiet, opts)

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(topics) command returns the list of available FunctionAdvisor topics.

• 

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.

analytic_extension

asymptotic_expansion

branch_cuts

branch_points

calling_sequence

class_members

classify_function

DE

definition

describe

differentiation_rule

display

function_classes

identities

integral_form

known_functions

relate

series

singularities

special_values

specialize

sum_form

synonyms

topics

  

The FunctionAdvisor command provides information on the following mathematical functions.

abs

AiryAi

AiryBi

AngerJ

arccos

arccosh

arccot

arccoth

arccsc

arccsch

arcsec

arcsech

arcsin

arcsinh

arctan

arctanh

argument

BellB

bernoulli

BesselI

BesselJ

BesselK

BesselY

Beta

binomial

ChebyshevT

ChebyshevU

Chi

Ci

CompleteBellB

cos

cosh

cot

coth

CoulombF

csc

csch

csgn

CylinderD

CylinderU

CylinderV

dawson

dilog

Dirac

doublefactorial

Ei

EllipticCE

EllipticCK

EllipticCPi

EllipticE

EllipticF

EllipticK

EllipticModulus

EllipticNome

EllipticPi

erf

erfc

erfi

euler

exp

factorial

FresnelC

Fresnelf

Fresnelg

FresnelS

GAMMA

GaussAGM

GegenbauerC

HankelH1

HankelH2

harmonic

Heaviside

HermiteH

HeunB

HeunBPrime

HeunC

HeunCPrime

HeunD

HeunDPrime

HeunG

HeunGPrime

HeunT

HeunTPrime

hypergeom

Hypergeom

Im

IncompleteBellB

InverseJacobiAM

InverseJacobiCD

InverseJacobiCN

InverseJacobiCS

InverseJacobiDC

InverseJacobiDN

InverseJacobiDS

InverseJacobiNC

InverseJacobiND

InverseJacobiNS

InverseJacobiSC

InverseJacobiSD

InverseJacobiSN

JacobiAM

JacobiCD

JacobiCN

JacobiCS

JacobiDC

JacobiDN

JacobiDS

JacobiNC

JacobiND

JacobiNS

JacobiP

JacobiSC

JacobiSD

JacobiSN

JacobiTheta1

JacobiTheta2

JacobiTheta3

JacobiTheta4

JacobiZeta

KelvinBei

KelvinBer

KelvinHei

KelvinHer

KelvinKei

KelvinKer

KummerM

KummerU

LaguerreL

LambertW

LegendreP

LegendreQ

LerchPhi

Li

ln

lnGAMMA

log

LommelS1

LommelS2

MathieuA

MathieuB

MathieuC

MathieuCE

MathieuCEPrime

MathieuCPrime

MathieuExponent

MathieuFloquet

MathieuFloquetPrime

MathieuS

MathieuSE

MathieuSEPrime

MathieuSPrime

MeijerG

pochhammer

polylog

Psi

Re

sec

sech

Shi

Si

signum

sin

sinh

SphericalY

Ssi

Stirling1

Stirling2

StruveH

StruveL

tan

tanh

WeberE

WeierstrassP

WeierstrassPPrime

WeierstrassSigma

WeierstrassZeta

WhittakerM

WhittakerW

Wrightomega

Zeta

  

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.

`0F1`

`1F1`

`2F1`

Airy

arctrig

arctrigh

Bessel_related

Chebyshev

Cylinder

Ei_related

elementary

Elliptic_doubly_periodic

Elliptic_related

erf_related

GAMMA_related

Hankel

hypergeometric

Jacobi_related

Kelvin

Kummer

Legendre

Lommel

orthogonal_polynomials

Other

Psi_related

Struve_related

trig

trigh

Weierstrass_related

Whittaker

Zeta_related

 

• 

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.

  

You do not have to remember the exact Maple name of each mathematical function or the FunctionAdvisor topic. However, to avoid these messages and all FunctionAdvisor verbosity, specify the optional argument quiet when calling the FunctionAdvisor command from another routine.

Examples

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

FunctionAdvisor

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.

FunctionAdvisortopics

The topics on which information is available are:

DE,analytic_extension,asymptotic_expansion,branch_cuts,branch_points,calling_sequence,class_members,classify_function,definition,describe,differentiation_rule,function_classes,identities,integral_form,known_functions,periodicity,relate,required_assumptions,series,singularities,special_values,specialize,sum_form,symmetries,synonyms

(1)

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

FunctionAdvisorfunction_classes,quiet

trig,trigh,arctrig,arctrigh,elementary,GAMMA_related,Psi_related,Kelvin,Airy,Hankel,Bessel_related,0F1,orthogonal_polynomials,Ei_related,erf_related,Kummer,Whittaker,Cylinder,1F1,Elliptic_related,Legendre,Chebyshev,2F1,Lommel,Struve_related,hypergeometric,Jacobi_related,InverseJacobi_related,Elliptic_doubly_periodic,Weierstrass_related,Zeta_related,complex_components,piecewise_related,Other,Bell,Heun,trigall,arctrigall,integral_transforms

(2)

The type of information that the FunctionAdvisor command returns general information, for example, "the Maple names for the Bessel functions",

FunctionAdvisorbess

* Partial match of "bess" against topic "Bessel_related".
The 14 functions in the "Bessel_related" class are:

AiryAi,AiryBi,BesselI,BesselJ,BesselK,BesselY,HankelH1,HankelH2,KelvinBei,KelvinBer,KelvinHei,KelvinHer,KelvinKei,KelvinKer

(3)

FunctionAdvisordescribe,BesselK

BesselK=Modified Bessel function of the second kind

(4)

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

FunctionAdvisorsum_form,tan

tanz&equals;_k1&equals;1&infin;bernoulli2_k11_k1z1&plus;2_k14_k116_k1&Gamma;2_k1&plus;1&comma;Andz<12&pi;

(5)

FunctionAdvisorintegral_form&comma;&Beta;

&Beta;x&comma;y&equals;&int;01_k1x11_k1y1&DifferentialD;_k1&comma;0<xAnd0<y

(6)

sp_eq:=FunctionAdvisorspecialize&comma;HermiteH&comma;KummerU

sp_eq:=HermiteHa&comma;z&equals;2aKummerU12a&comma;12&comma;z2&comma;And0<&real;zorz=0And0<z

(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

hassp_eq&comma;a&comma;z

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 EllipticFz&comma;k.

FunctionAdvisorsyntax&comma;EllipticF

EllipticFz&comma;k

(9)

EF_and_DE:=FunctionAdvisorDE&comma;EllipticFz&comma;k

EF_and_DE:=fz&comma;k&equals;EllipticFz&comma;k&comma;2k2fz&comma;k&equals;13k4z2&plus;z2&plus;3k2kfz&comma;kk5z2&plus;z21k3&plus;k&plus;z3zzfz&comma;kk4k2z2k2&plus;1&plus;k2z2&plus;1fz&comma;kk4z2k2z2k2&plus;1&comma;2zkfz&comma;k&equals;zfz&comma;kkz2k2z21&comma;2z2fz&comma;k&equals;2k2z3&plus;k2&plus;1zzfz&comma;k1&plus;k2z4&plus;k21z2

(10)

map2has&comma;EF_and_DE&comma;z&comma;k

true&comma;true

(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.

pdetestEF_and_DE1&comma;EF_and_DE2

0&comma;0&comma;0

(12)

EF_and_DE1

fz&comma;k&equals;EllipticFz&comma;k

(13)

convertEF_and_DE1&comma;Int

fz&comma;k&equals;&int;0z1_&alpha;12&plus;1_&alpha;12k2&plus;1&DifferentialD;_&alpha;1

(14)

Use the FunctionAdvisor command to return a table of information for the arccot function. (See also FunctionAdvisor/display.)

FunctionAdvisorarccot&comma;quiet&colon;

FunctionAdvisordisplay&comma;arccot

arccot belongs to the subclass "arctrig" of the class "elementary" and so, in principle, it can be related to various of the 26 functions of those classes - see FunctionAdvisor( "arctrig" ); and FunctionAdvisor( "elementary" );

describe&equals;arccot&equals;inverse cotangent function

classify_function&equals;arctrig&comma;elementary

definition&equals;arccotz&equals;12&pi;12Iln1Izln1&plus;Iz&comma;with no restrictions on z

symmetries&equals;arccotz&equals;&pi;arccotz&comma;arccotz&conjugate0;&equals;arccotz&conjugate0;&comma;notz&in;ComplexRange&infin;I&comma;Iorz&in;ComplexRangeI&comma;&infin;I

periodicity&equals;No periodicity

singularities&equals;arccotz&comma;z&equals;&infin;&plus;&infin;I

branch_points&equals;arccotz&comma;z&in;I&comma;I

branch_cuts&equals;arccotz&comma;z&in;ComplexRange&infin;I&comma;I&comma;z&in;ComplexRangeI&comma;&infin;I

special_values&equals;arccot1&equals;34&pi;&comma;arccot133&equals;23&pi;&comma;arccot3&equals;56&pi;&comma;arccot0&equals;12&pi;&comma;arccot3&equals;16&pi;&comma;arccot133&equals;13&pi;&comma;arccot1&equals;14&pi;&comma;arccot&infin;&equals;0&comma;arccot&infin;&equals;&pi;

identities&equals;cotarccotx&equals;x&comma;cotarccotx&plus;arccoty&equals;xy1x&plus;y

sum_form&equals;arccotz&equals;_k1&equals;0&infin;zIz_k1&plus;Iz_k12_k1&plus;2&plus;12&pi;&comma;Andz<1

series&equals;seriesarccotz&comma;z&comma;4&equals;12&pi;z&plus;13z3&plus;Oz5

asymptotic_expansion&equals;asymptarccotz&comma;z&comma;4&equals;1z13z3&plus;O1z5

integral_form&equals;arccotz&equals;&int;1&plus;Iz1Iz12I_k1&DifferentialD;_k1&plus;12&pi;&comma;with no restrictions on z

differentiation_rule&equals;&DifferentialD;&DifferentialD;zarccotz&equals;1z2&plus;1

DE&equals;fz&equals;arccotz&comma;&DifferentialD;&DifferentialD;zfz&equals;1z2&plus;1

(15)

The relation between all elementary functions and the pFq hypergeometric function:

fncs:=FunctionAdvisorelementary&comma;quiet

fncs:=arccos&comma;arccosh&comma;arccot&comma;arccoth&comma;arccsc&comma;arccsch&comma;arcsec&comma;arcsech&comma;arcsin&comma;arcsinh&comma;arctan&comma;arctanh&comma;cos&comma;cosh&comma;cot&comma;coth&comma;csc&comma;csch&comma;exp&comma;ln&comma;sec&comma;sech&comma;sin&comma;sinh&comma;tan&comma;tanh

(16)

map2FunctionAdvisor&comma;relate&comma;fncs&comma;hypergeom

arccosz&equals;12&pi;zhypergeom12&comma;12&comma;32&comma;z2&comma;arccoshz&equals;1&plus;z22zhypergeom12&comma;12&comma;32&comma;z2&pi;2&plus;2z&comma;arccotz&equals;12&pi;zhypergeom12&comma;1&comma;32&comma;z2&comma;arccothz&equals;&pi;z12&plus;2zhypergeom12&comma;1&comma;32&comma;z2z12z2&comma;arccscz&equals;hypergeom12&comma;12&comma;32&comma;1z2z&comma;arccschz&equals;hypergeom12&comma;12&comma;32&comma;1z2z&comma;arcsecz&equals;12&pi;hypergeom12&comma;12&comma;32&comma;1z2z&comma;arcsechz&equals;z12z2z&pi;2hypergeom12&comma;12&comma;32&comma;1z22z2&comma;arcsinz&equals;zhypergeom12&comma;12&comma;32&comma;z2&comma;arcsinhz&equals;zhypergeom12&comma;12&comma;32&comma;z2&comma;arctanz&equals;zhypergeom12&comma;1&comma;32&comma;z2&comma;arctanhz&equals;zhypergeom12&comma;1&comma;32&comma;z2&comma;cosz&equals;hypergeom&comma;12&comma;14z2&comma;coshz&equals;hypergeom&comma;12&comma;14z2&comma;cotz&equals;hypergeom&comma;12&comma;14z2zhypergeom&comma;32&comma;14z2&comma;cothz&equals;hypergeom&comma;12&comma;14z2zhypergeom&comma;32&comma;14z2&comma;cscz&equals;1zhypergeom&comma;32&comma;14z2&comma;cschz&equals;1zhypergeom&comma;32&comma;14z2&comma;&ExponentialE;z&equals;hypergeom&comma;&comma;z&comma;lnz&equals;z1hypergeom1&comma;1&comma;2&comma;z&plus;1&comma;secz&equals;1hypergeom&comma;12&comma;