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FunctionAdvisor/relate

relate the two given mathematical functions

	Calling Sequence
	FunctionAdvisor(relate, math_function_1, math_function_2)

Parameters

relate	-	literal name; 'relate'
math_function_1	-	Maple name of first mathematical function
math_function_2	-	Maple name of second mathematical function

Description

•	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).

Examples

>	$FunctionAdvisor (relate, \sin, WhittakerM)$

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

(1)

>	$FunctionAdvisor (relate, KummerU, BesselK)$

$BesselK (a, z) = \frac{\sqrt{π} {(2 z)}^{a} KummerU (a + \frac{1}{2}, 2 a + 1, 2 z)}{{&ExponentialE;}^{z}}$

(2)

>	$FunctionAdvisor (relate, HermiteH, \erf)$

$\erf (z) = - \frac{2 HermiteH (−1, z)}{\sqrt{π} {&ExponentialE;}^{z^{2}}} + 1$

(3)

>	$FunctionAdvisor (relate, HermiteH, LaguerreL)$

$HermiteH (a, z) = 2^{a} \sqrt{π} (\frac{LaguerreL (\frac{a}{2}, - \frac{1}{2}, z^{2})}{(\binom{- \frac{1}{2} + \frac{a}{2}}{\frac{a}{2}}) Γ (\frac{1}{2} - \frac{a}{2})} - \frac{2 z LaguerreL (- \frac{1}{2} + \frac{a}{2}, \frac{1}{2}, z^{2})}{(\binom{\frac{a}{2}}{- \frac{1}{2} + \frac{a}{2}}) Γ (- \frac{a}{2})})$

(4)

Notice that a relation between KummerU and HermiteH does not exist for arbitrary values of their parameters.

>	$FunctionAdvisor (relate, KummerU, Hermite)$

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

However, that relation does exist for a specialization of the function parameters

>	$FunctionAdvisor (specialize, KummerU, HermiteH)$

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

(5)

and also the converse.

>	$ex1 ≔ FunctionAdvisor (specialize, HermiteH, KummerU)$

$ex1 ≔ [HermiteH (a, z) = 2^{a} KummerU (- \frac{a}{2}, \frac{1}{2}, z^{2}), 0 < ℜ (z) \lor (ℜ (z) = 0 \land 0 < ℑ (z))]$

(6)

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

>	$depends ([ex1], a), depends ([ex1], b), depends ([ex1], z)$

$false, false, false$

(7)

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

>	$ex2 ≔ FunctionAdvisor (specialize, HermiteH (a, z), KummerU)$

$ex2 ≔ [HermiteH (a, z) = 2^{a} KummerU (- \frac{a}{2}, \frac{1}{2}, z^{2}), 0 < ℜ (z) \lor (ℜ (z) = 0 \land 0 < ℑ (z))]$

(8)

>	$depends ([ex2], a), depends ([ex2], b), depends ([ex2], z)$

$true, false, true$

(9)

You can view the default hierarchy used when calling the FunctionAdvisor command with the keyword relate as first argument using the following

>	$`convert/to_special_function/hierarchy` [function]$

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

(10)

where functions to the left are selected before functions to the right when constructing the left-hand side of the output.

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