# Alternative closed forms for digamma expressions

What commands should I use such that Mathematica / W|A express

$$\psi(1+i)+\psi(1-i)$$ exclusively in terms of trigonometric functions?

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Maybe you could help by letting us know which Mathematica command(s) gives "digamma"? – bill s Feb 1 '14 at 17:49
What is i in your function and how does it relate to x? (Hint: show code that you have tried). – bill s Feb 1 '14 at 17:54
I was not the downvoter, but your comment is quite arrogant given how poorly you wrote your question. – bill s Feb 1 '14 at 18:19
For rational arguments, en.wikipedia.org/wiki/… may be helpful. – Eckhard Feb 1 '14 at 18:31
So you asked to be deleted yesterday and then posted a question today? Asking for your account to be deleted is OK. Defacing stackexchange is not OK. Here are the instructions: meta.stackoverflow.com/help/deleting-account – bill s Feb 1 '14 at 18:56

## 1 Answer

How about a series expansion?

Chop@N@Series[PolyGamma[0, x], {x, 0, 5}]


Or try FullSimplify:

FullSimplify[PolyGamma[1 - x] - PolyGamma[x]]
π Cot[π x]

FullSimplify[PolyGamma[1 - x] - PolyGamma[1 + x]]
-(1/x) + π Cot[π x]


With the plus I get the answer in terms of HarmonicNumbers rather than Trig functions:

N@FullSimplify[PolyGamma[1 - x] + PolyGamma[1 + x]]
-1.15443 + HarmonicNumber[-1. x] + HarmonicNumber[x]

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Did you know that $\psi(1+i)-\psi(1-i)=i(\pi \coth(\pi)-1)$? I wonder if something similar exists when using addition insted of subtraction. The point is: how can I force Mathematica give me such alternative forms? – A_math_ninja Feb 1 '14 at 18:03
@Chris'ssis: Executing FullSimplify[PolyGamma[1 + x] + PolyGamma[1 - x]] gives $H_{-x}+H_x-2 \gamma$ where $H$ is the analytic continuation of the Harmonic number series and where $\gamma$ is the Euler-Gamma constant. Are you sure there exists a trig-function equivalent? If there is, then Mathematica probably can't do it. Also, please stop screaming and whining like an 8-year old in ALL CAPS. It makes you look like a baby. – DumpsterDoofus Feb 1 '14 at 19:32
I see. Have you tried quitting the kernel and restarting and running it again? I am using version 9 and I get imagebin.org/290343. If after restarting and trying again you still get $-\infty$, then it's probably an error in version 8. Mathematica does have glitches sometimes. And I never said you lied. I said that you didn't offer proof, which made it impossible for people to find what is going wrong, which is why everyone downvoted you. I know you're not a liar, and I know you're a very good mathematician. But you need to write your questions clearly to get good answers. – DumpsterDoofus Feb 1 '14 at 19:39
@DumpsterDoofus everything is perfect here. Mathematica simply makes a big mistake. Yesterday morning W|A made the same mistake again and again ... after my post things were fixed. – A_math_ninja Feb 1 '14 at 19:40
Do you have a screenshot of W|A giving this glitch? – DumpsterDoofus Feb 1 '14 at 19:41