The FunctionAdvisor(relate, math_function_1, math_function_2) command attempts to relate the functions using the internal Maple algorithms. For more information, see convert[to_special_function].

By default, the output has the simpler or more standard function on the left-hand side. This topic is similar to FunctionAdvisor/specialize, which allows you to specify which function is expressed in terms of the other one. This typically requires additional constraints in the function parameter(s) domain(s).

$\mathrm{FunctionAdvisor}\left(\mathrm{relate}\,\sin \,\mathrm{WhittakerM}\right)$

$\sin \left(z\right)=-\frac{\mathrm{I}}{2}\mathrm{WhittakerM}\left(0\,\frac{1}{2}\,2\mathrm{I}z\right)$

$\mathrm{FunctionAdvisor}\left(\mathrm{relate}\,\mathrm{KummerU}\,\mathrm{BesselK}\right)$

$\mathrm{BesselK}\left(a\,z\right)=\frac{\sqrt{\mathrm{\pi }}{\left(2z\right)}^a\mathrm{KummerU}\left(a+\frac{1}{2}\,2a+1\,2z\right)}{{\ⅇ}^z}$

$\mathrm{FunctionAdvisor}\left(\mathrm{relate}\,\mathrm{HermiteH}\,\mathrm{erf}\right)$

$\mathrm{erf}\left(z\right)=-\frac{2\mathrm{HermiteH}\left(\mathrm{-1}\,z\right)}{\sqrt{\mathrm{\pi }}{\ⅇ}^{z^2}}+1$

$\mathrm{FunctionAdvisor}\left(\mathrm{relate}\,\mathrm{HermiteH}\,\mathrm{LaguerreL}\right)$

$\mathrm{HermiteH}\left(a\,z\right)=2^a\sqrt{\mathrm{\pi }}\left(\frac{\mathrm{LaguerreL}\left(\frac{a}{2}\,-\frac{1}{2}\,z^2\right)}{\left(\genfrac{}{}{0ex}{}{-\frac{1}{2}+\frac{a}{2}}{\frac{a}{2}}\right)\mathrm{\Gamma }\left(\frac{1}{2}-\frac{a}{2}\right)}-\frac{2z\mathrm{LaguerreL}\left(-\frac{1}{2}+\frac{a}{2}\,\frac{1}{2}\,z^2\right)}{\left(\genfrac{}{}{0ex}{}{\frac{a}{2}}{-\frac{1}{2}+\frac{a}{2}}\right)\mathrm{\Gamma }\left(-\frac{a}{2}\right)}\right)$

$\mathrm{FunctionAdvisor}\left(\mathrm{relate}\,\mathrm{KummerU}\,\mathrm{Hermite}\right)$

* Partial match of Hermite against topic HermiteH.
Unable to establish a relation between KummerU and HermiteH

$\mathrm{FunctionAdvisor}\left(\mathrm{specialize}\,\mathrm{KummerU}\,\mathrm{HermiteH}\right)$

$\left[\mathrm{KummerU}\left(a\,b\,z\right)=\mathrm{HermiteH}\left(-2a\,\sqrt{z}\right)2^{2a}\,b=\frac{1}{2}\right],\left[\mathrm{KummerU}\left(a\,b\,z\right)=\frac{\mathrm{HermiteH}\left(-2a+1\,\sqrt{z}\right)2^{2a}}{2\sqrt{z}}\,b=\frac{3}{2}\right]$

$\mathrm{ex1}≔\mathrm{FunctionAdvisor}\left(\mathrm{specialize}\,\mathrm{HermiteH}\,\mathrm{KummerU}\right)$

$\mathrm{ex1}≔\left[\mathrm{HermiteH}\left(a\&comma;z\right)=2^a\mathrm{KummerU}\left(-\frac{a}{2}\&comma;\frac{1}{2}\&comma;z^2\right)\&comma;0<\mathrm{\Re }\left(z\right)\vee \left(\mathrm{\Re }\left(z\right)=0\wedge 0<\mathrm{\Im }\left(z\right)\right)\right]$

$\mathrm{depends}\left(\left[\mathrm{ex1}\right]\&comma;a\right),\mathrm{depends}\left(\left[\mathrm{ex1}\right]\&comma;b\right),\mathrm{depends}\left(\left[\mathrm{ex1}\right]\&comma;z\right)$

$\mathrm{false},\mathrm{false},\mathrm{false}$

$\mathrm{ex2}≔\mathrm{FunctionAdvisor}\left(\mathrm{specialize}\&comma;\mathrm{HermiteH}\left(a\&comma;z\right)\&comma;\mathrm{KummerU}\right)$

$\mathrm{ex2}≔\left[\mathrm{HermiteH}\left(a\&comma;z\right)=2^a\mathrm{KummerU}\left(-\frac{a}{2}\&comma;\frac{1}{2}\&comma;z^2\right)\&comma;0<\mathrm{\Re }\left(z\right)\vee \left(\mathrm{\Re }\left(z\right)=0\wedge 0<\mathrm{\Im }\left(z\right)\right)\right]$

$\mathrm{depends}\left(\left[\mathrm{ex2}\right]\&comma;a\right),\mathrm{depends}\left(\left[\mathrm{ex2}\right]\&comma;b\right),\mathrm{depends}\left(\left[\mathrm{ex2}\right]\&comma;z\right)$

$\mathrm{true},\mathrm{false},\mathrm{true}$

$\mathrm{`convert/to\_special\_function/hierarchy`}\left[''function''\right]$

$\left[\sin \&comma;\cos \&comma;\tan \&comma;\csc \&comma;\sec \&comma;\cot \&comma;\sinh \&comma;\cosh \&comma;\tanh \&comma;\mathrm{csch}\&comma;\mathrm{sech}\&comma;\coth \&comma;\arcsin \&comma;\arccos \&comma;\arctan \&comma;\mathrm{arccsc}\&comma;\mathrm{arcsec}\&comma;\mathrm{arccot}\&comma;\mathrm{arcsinh}\&comma;\mathrm{arccosh}\&comma;\mathrm{arctanh}\&comma;\mathrm{arccsch}\&comma;\mathrm{arcsech}\&comma;\mathrm{arccoth}\&comma;\exp \&comma;\ln \&comma;\mathrm{\Gamma }\&comma;\mathrm{lnGAMMA}\&comma;\mathrm{factorial}\&comma;\mathrm{binomial}\&comma;\mathrm{multinomial}\&comma;\mathrm{{\rm B}}\&comma;\mathrm{pochhammer}\&comma;\mathrm{doublefactorial}\&comma;\mathrm{\Psi }\&comma;\mathrm{harmonic}\&comma;\mathrm{KelvinBer}\&comma;\mathrm{KelvinBei}\&comma;\mathrm{KelvinKer}\&comma;\mathrm{KelvinKei}\&comma;\mathrm{KelvinHer}\&comma;\mathrm{KelvinHei}\&comma;\mathrm{AiryAi}\&comma;\mathrm{AiryBi}\&comma;\mathrm{HankelH1}\&comma;\mathrm{HankelH2}\&comma;\mathrm{BesselI}\&comma;\mathrm{BesselJ}\&comma;\mathrm{BesselK}\&comma;\mathrm{BesselY}\&comma;\mathrm{GegenbauerC}\&comma;\mathrm{HermiteH}\&comma;\mathrm{LaguerreL}\&comma;\mathrm{LegendreP}\&comma;\mathrm{ChebyshevT}\&comma;\mathrm{ChebyshevU}\&comma;\mathrm{JacobiP}\&comma;\mathrm{Ei}\&comma;\mathrm{Li}\&comma;\mathrm{Si}\&comma;\mathrm{Shi}\&comma;\mathrm{Ci}\&comma;\mathrm{Chi}\&comma;\mathrm{Ssi}\&comma;\mathrm{erf}\&comma;\mathrm{erfc}\&comma;\mathrm{erfi}\&comma;\mathrm{FresnelC}\&comma;\mathrm{FresnelS}\&comma;\mathrm{Fresnelf}\&comma;\mathrm{Fresnelg}\&comma;\mathrm{dawson}\&comma;\mathrm{KummerM}\&comma;\mathrm{KummerU}\&comma;\mathrm{WhittakerM}\&comma;\mathrm{WhittakerW}\&comma;\mathrm{CylinderD}\&comma;\mathrm{CylinderU}\&comma;\mathrm{CylinderV}\&comma;\mathrm{CoulombF}\&comma;\mathrm{EllipticE}\&comma;\mathrm{EllipticCE}\&comma;\mathrm{EllipticF}\&comma;\mathrm{InverseJacobiSN}\&comma;\mathrm{InverseJacobiAM}\&comma;\mathrm{EllipticK}\&comma;\mathrm{EllipticCK}\&comma;\mathrm{EllipticPi}\&comma;\mathrm{EllipticCPi}\&comma;\mathrm{EllipticModulus}\&comma;\mathrm{EllipticNome}\&comma;\mathrm{LegendreQ}\&comma;\mathrm{SphericalY}\&comma;\mathrm{LerchPhi}\&comma;\mathrm{GaussAGM}\&comma;\mathrm{LommelS1}\&comma;\mathrm{MeijerG}\&comma;\mathrm{hypergeom}\&comma;\mathrm{Hypergeom}\&comma;\mathrm{AppellF1}\&comma;\mathrm{AppellF2}\&comma;\mathrm{AppellF3}\&comma;\mathrm{AppellF4}\&comma;\mathrm{Sum}\&comma;\mathrm{sum}\&comma;\mathrm{LommelS2}\&comma;\mathrm{StruveH}\&comma;\mathrm{StruveL}\&comma;\mathrm{AngerJ}\&comma;\mathrm{WeberE}\&comma;\mathrm{dilog}\&comma;\mathrm{polylog}\&comma;\mathrm{JacobiAM}\&comma;\mathrm{JacobiSN}\&comma;\mathrm{JacobiCN}\&comma;\mathrm{JacobiDN}\&comma;\mathrm{JacobiNS}\&comma;\mathrm{JacobiNC}\&comma;\mathrm{JacobiND}\&comma;\mathrm{JacobiSC}\&comma;\mathrm{JacobiCS}\&comma;\mathrm{JacobiSD}\&comma;\mathrm{JacobiDS}\&comma;\mathrm{JacobiCD}\&comma;\mathrm{JacobiDC}\&comma;\mathrm{JacobiTheta1}\&comma;\mathrm{JacobiTheta2}\&comma;\mathrm{JacobiTheta3}\&comma;\mathrm{JacobiTheta4}\&comma;\mathrm{JacobiZeta}\&comma;\mathrm{InverseJacobiCN}\&comma;\mathrm{InverseJacobiDN}\&comma;\mathrm{InverseJacobiNS}\&comma;\mathrm{InverseJacobiNC}\&comma;\mathrm{InverseJacobiND}\&comma;\mathrm{InverseJacobiSC}\&comma;\mathrm{InverseJacobiCS}\&comma;\mathrm{InverseJacobiSD}\&comma;\mathrm{InverseJacobiDS}\&comma;\mathrm{InverseJacobiCD}\&comma;\mathrm{InverseJacobiDC}\&comma;\mathrm{WeierstrassP}\&comma;\mathrm{WeierstrassPPrime}\&comma;\mathrm{WeierstrassZeta}\&comma;\mathrm{WeierstrassSigma}\&comma;\mathrm{{\rm Z}}\&comma;\mathrm{abs}\&comma;\mathrm{conjugate}\&comma;\mathrm{\Re }\&comma;\mathrm{\Im }\&comma;\mathrm{argument}\&comma;\mathrm{signum}\&comma;\mathrm{Dirac}\&comma;\mathrm{Heaviside}\&comma;\max \&comma;\min \&comma;\mathrm{piecewise}\&comma;\mathrm{LambertW}\&comma;\mathrm{Wrightomega}\&comma;\mathrm{unwindK}\&comma;\mathrm{euler}\&comma;\mathrm{bernoulli}\&comma;\mathrm{Stirling1}\&comma;\mathrm{Stirling2}\&comma;\mathrm{BellB}\&comma;\mathrm{CompleteBellB}\&comma;\mathrm{IncompleteBellB}\&comma;\mathrm{HeunT}\&comma;\mathrm{HeunB}\&comma;\mathrm{HeunD}\&comma;\mathrm{HeunC}\&comma;\mathrm{HeunG}\&comma;\mathrm{HeunTPrime}\&comma;\mathrm{HeunBPrime}\&comma;\mathrm{HeunDPrime}\&comma;\mathrm{HeunCPrime}\&comma;\mathrm{HeunGPrime}\&comma;\mathrm{MathieuC}\&comma;\mathrm{MathieuCPrime}\&comma;\mathrm{MathieuS}\&comma;\mathrm{MathieuSPrime}\&comma;\mathrm{MathieuCE}\&comma;\mathrm{MathieuCEPrime}\&comma;\mathrm{MathieuSE}\&comma;\mathrm{MathieuSEPrime}\&comma;\mathrm{MathieuFloquet}\&comma;\mathrm{MathieuFloquetPrime}\&comma;\mathrm{MathieuA}\&comma;\mathrm{MathieuB}\&comma;\mathrm{MathieuExponent}\&comma;\mathrm{GeneralizedPolylog}\&comma;\mathrm{MultiPolylog}\&comma;\mathrm{MultiZeta}\&comma;\mathrm{NielsenPolylog}\&comma;\mathrm{fourier}\&comma;\mathrm{invfourier}\&comma;\mathrm{fouriercos}\&comma;\mathrm{fouriersin}\&comma;\mathrm{hankel}\&comma;\mathrm{hilbert}\&comma;\mathrm{invhilbert}\&comma;\mathrm{laplace}\&comma;\mathrm{invlaplace}\&comma;\mathrm{mellin}\&comma;\mathrm{invmellin}\right]$