10
$\begingroup$

Is there any way that one can describe all the roots of the incomplete gamma function $\Gamma(n,z)$, for $n\in \mathbb{N}$, analytically?

$\endgroup$
24
$\begingroup$

(This was supposed to be another comment to Raymond's answer, but it got too long.)

As already mentioned by Raymond, we have the relationship (using slightly different notation)

$$\Gamma(n,z)= (n-1)! \exp(-z) e_{n-1}(z)$$

where

$$e_n(z)=\sum_{k=0}^n \frac{z^k}{k!}$$

the partial sums of the Maclaurin series for $\exp(z)$, is sometimes called the exponential sum function. Though a general closed form for the roots of $e_n(z)$ is not known, the distribution of the roots of the polynomial $e_n(nz)$ in the complex plane is pretty well studied:

scaled roots of exponential sum functions

In particular, Gábor Szegő (1924) and Jean Dieudonné (1935) both showed that the roots of the scaled exponential sum function $e_n(nz)$ approach the portion of the curve $|z\exp(1-z)|=1$ (now often referred to as the Szegő curve) within the unit disk as $n\to\infty$. In two papers, Carpenter, Varga, and Waldvogel study the asymptotics of the zeros of $e_n(nz)$. Other papers of interest include those by Buckholtz, Newman and Rivlin (with correction), Conrey and Ghosh, Pritsker and Varga, Walker, and Zemyan. (These are the ones I've read; I'm sure there are other nice papers I've missed.)

$\endgroup$
  • 4
    $\begingroup$ Thanks for the interesting links and nice illustration! I'll just add that the second paper of Varga and Carpenter 'Asymptotics for the zeros of the partial sums of e^z. II' (and many others on the same subject and the related 'partial sums of $\cos(z)\ $ and $\sin(z)\ $') are available at Varga's homepage. Cheers, $\endgroup$ – Raymond Manzoni Feb 15 '12 at 20:11
  • $\begingroup$ Very nice, Thanks so much for the graph and references. $\endgroup$ – Keivan Feb 16 '12 at 2:00
  • $\begingroup$ This may also be interesting. $\endgroup$ – Jean-Claude Arbaut Nov 5 '15 at 13:44
4
$\begingroup$

With the usual definition we have : $$\Gamma(n,x)=\int_x^{\infty} t^{n-1} e^{-t} dt$$

After repeatedly using integration by parts we may get this formula : $$\Gamma(n,x)= (n-1)! e^{-x} \sum_{k=0}^{n-1} \frac{x^k}{k!}$$ so that you are asking for the zeros of the polynomials : $$P_n(x)=\sum_{k=0}^{n-1} \frac{x^k}{k!}$$ Let's search the first solutions :

$ \begin{array} {rcc} n & P_n(x) & \mathrm{zeros} \\ \hline \\ 1 & 1 & \emptyset \\ 2 & 1+x & \{-1\} \\ 3 & 1+x+\frac{x^2}{2!} & \{-1-i,-1+i\} \\ 4 & 1+x+\frac{x^2}{2!}+\frac{x^3}{3!} & \{\sqrt[3]{\sqrt{2}-1}-\frac1{\sqrt[3]{\sqrt{2}-1}}-1,..,.. \} \\ \end{array} $

I'll stop here because the algebraic expressions of the zeros are becoming rather long and complicated... (search others with WolframAlpha)
Or did you hope something simpler?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.