I'm trying to evaluate the Gamma function in Bessel's equation for $v=1/4$.

What is $\Gamma(m+5/4)$, or when solving, what is $\Gamma(1/4)$?

  • $\begingroup$ $$\Gamma\left(m+\dfrac{5}{4}\right)=(m+\dfrac{1}{4})\Gamma\left(m+\dfrac{1}{4}\right)$$ $\endgroup$ – Nosrati Sep 6 '17 at 4:55
  • $\begingroup$ what does the Γ(m+1/4) term equal? $\endgroup$ – watdoing Sep 6 '17 at 4:58
  • $\begingroup$ See general form $x\Gamma(x)=\Gamma(x+1)$ $\endgroup$ – Nosrati Sep 6 '17 at 5:00
  • $\begingroup$ What do you mean by "solve"? $\Gamma(1/4) \approx 3.625609909$. It is a transcendental constant, which can be related to some others (e.g. see MathWorld), but I don't think there's any expression for it that I'd call "simpler" than $\Gamma(1/4)$. $\endgroup$ – Robert Israel Sep 6 '17 at 5:11

$$\Gamma\left(m+\frac{1}{4}\right)=\Gamma\left(\frac{1}{4}\right)\prod _{k=1}^m\left(k-\frac{3}{4}\right)$$ $\Gamma\left(\frac{1}{4}\right)$ is a remarkable constant. $$\Gamma\left(\frac{1}{4}\right)\simeq 3.62560990822190831...$$ In the book "An Atlas of Functions",J.Spannier, K.B.Oldham, this constant is noted U and called the "ubiquitous constant". It appears throughout many relationships involving special functions. One most notable is : $$\Gamma\left(\frac{1}{4}\right)=2\sqrt{\sqrt{\pi}\text{K}\left(1/2\right)}$$ where K$(x)$ is the complete elliptic integral of the first kind.

An efficient iterative process to evaluate $\Gamma\left(\frac{1}{4}\right)$ is : $$\Gamma\left(\frac{1}{4}\right)=\text{Arithmetic-Geometric Mean of }1 \text{ and }\frac{1}{\sqrt{2}}$$

The arithmetic-geometric mean of two numbers $A_0\:,\:B_0$ is approached by repeating : $A_{k+1}=(A_k+B_k)/2$ and $B_{k+1}=\sqrt{A_kB_k}$.

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    $\begingroup$ What is called the "common mean" in this answer is more conventionally called the "arithmetic-geometric mean", or the AGM. $\endgroup$ – J. M. is a poor mathematician Sep 6 '17 at 6:33
  • $\begingroup$ Yes, "arithmetic-geometric mean" is more popular. I will change the notation in my answer. Thanks you. $\endgroup$ – JJacquelin Sep 6 '17 at 6:48

Some properties of that value are gathered in the associated wikipedia article.


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