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In[1]:= TrueQ[Gamma[2 z] == (2^(2 z) Gamma[z] Gamma[z + 1/2])/(2 Sqrt[\[Pi]])]

Out[1]:= False

Can someone explain why the above is returning false?

http://mathworld.wolfram.com/LegendreDuplicationFormula.html

I don't know what to tag this as, so I chose simplifying expressions.

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    $\begingroup$ Developer`GammaSimplify[ Gamma[2 z] == (2^(2 z) Gamma[z] Gamma[z + 1/2])/(2 Sqrt[\[Pi]])] gives True. $\endgroup$
    – kglr
    Jun 7, 2019 at 9:18
  • $\begingroup$ so does Simplify`SimplifyGamma[ Gamma[2 z] == (2^(2 z) Gamma[z] Gamma[z + 1/2])/(2 Sqrt[\[Pi]])]. $\endgroup$
    – kglr
    Jun 7, 2019 at 9:20

1 Answer 1

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The argument of TrueQ doesn't evaluate to True. It does if you wrap FullSimplify around it:

TrueQ[Gamma[2 z] == (2^(2 z) Gamma[z] Gamma[z + 1/2])/(2 Sqrt[π]) // FullSimplify]

True

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  • $\begingroup$ What would be nice is if it did evaluate to True without having to do that.... But algebraic simplification is hard for computers, so I'll let it slide. $\endgroup$ Jun 7, 2019 at 9:17
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    $\begingroup$ @JackLam - Mathematica does not do much more than trivial simplification unless directed to do so. Consequently, your argument to TrueQ is not "explicitly True" and, as documented, TrueQ returns False. Once you FullSimplify, TrueQ is unnecessary since Gamma[2 z] == (2^(2 z) Gamma[z] Gamma[z+1/2])/(2 Sqrt[Pi])//FullSimplify evaluates to True. $\endgroup$
    – Bob Hanlon
    Jun 7, 2019 at 21:02

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