# The Lerch Phi Function and a possible Mathematica Bug [Solved]

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).

• It looks as though Mathematica starts at $k=1$. This will give you what you seek. – Simply Beautiful Art Dec 31 '16 at 13:57
• @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. – Von Neumann Dec 31 '16 at 13:59
• That is probably the case we are seeing. – Simply Beautiful Art Dec 31 '16 at 14:00
• Be careful not to mix up LerchPhi with HurwitzLerchPhi – polfosol Dec 31 '16 at 14:14
• See the first few lines under details – Simply Beautiful Art Dec 31 '16 at 14:16

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

• 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. – Simply Beautiful Art Dec 31 '16 at 14:19
• 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. – Von Neumann Dec 31 '16 at 14:21
• @AlanTuring There we go, that should be more obvious. – Simply Beautiful Art Dec 31 '16 at 14:22