0
$\begingroup$

The Lerch Phi Function is defined as

$$\Phi(-s, \alpha, \nu) = \sum_{k = 0}^{+\infty} \frac{(-s)^k}{(k+\nu)^{\alpha}}$$

Now, in my special case I have $\alpha = 1$, hence it does simply reduce to

$$\Phi(-s, 1, \nu) = \sum_{k = 0}^{+\infty} \frac{(-s)^k}{(k + \nu)}$$

Clearly if I put $\nu = 0$ the series shall be infinite.

But when I compute it with Mathematica (Wolfram Serious Mathematica software, not Online Alpha) it says that

$$\Phi(-s, 1, \nu) = -\ln(1+s)$$

How is this possible?

EDIT

There are also other problems related to that. For example setting $\nu = -1$ and Mathematica says

$$\Phi(-s, 1, -1) = 1+ s\ln(1+s)$$

Which is actually not true, since by its definition (both it starts from $k = 0$ or $k = 1$ there is a point in which the series is infinite).

$\endgroup$
  • $\begingroup$ It looks as though Mathematica starts at $k=1$. This will give you what you seek. $\endgroup$ – Simply Beautiful Art Dec 31 '16 at 13:57
  • $\begingroup$ @SimpleArt The problem is that the definition on Wolfram Documentations says it starts from $k = 0$. Sure $k = 1$ would solve the problem... If it's not a bug, then there is an error in the documentation form. $\endgroup$ – Von Neumann Dec 31 '16 at 13:59
  • $\begingroup$ That is probably the case we are seeing. $\endgroup$ – Simply Beautiful Art Dec 31 '16 at 14:00
  • 2
    $\begingroup$ Be careful not to mix up LerchPhi with HurwitzLerchPhi $\endgroup$ – polfosol Dec 31 '16 at 14:14
  • 1
    $\begingroup$ See the first few lines under details $\endgroup$ – Simply Beautiful Art Dec 31 '16 at 14:16
2
$\begingroup$

According to the reference, we have

$$\Phi(-s, \alpha, \nu) = \sum_{k = 0}^{+\infty} \frac{(-s)^k}{(k+\nu)^{\alpha}}$$

for $\Re(\nu)>0$. For $\Re(\nu)\le0$, we have

$$\Phi(-s, \alpha, \nu) = \sum_{k = 0}^{+\infty} \frac{(-s)^k}{[(k+\nu)^2]^{\alpha/2}}$$

where any term $k+\nu=0$ is eliminated.

References:


http://reference.wolfram.com/language/ref/LerchPhi.html

$\endgroup$
  • $\begingroup$ Huh, you know what's funny? I was reading about this function yesterday, and I actually noticed all this and was like "Wow, what a strange function!" only to forget and rediscover the next day. $\endgroup$ – Simply Beautiful Art Dec 31 '16 at 14:19
  • $\begingroup$ Haha! By the way, try to color the last link in RED, so that people will immediately read the strangeness of the Lerch phi! And thanks again. $\endgroup$ – Von Neumann Dec 31 '16 at 14:21
  • $\begingroup$ @AlanTuring There we go, that should be more obvious. $\endgroup$ – Simply Beautiful Art Dec 31 '16 at 14:22

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.