GAUSS DIGAMMA THEOREM
Today we will prove the famous Gauss digamma theorem, namely The proof is based on Jensen´s famous proof. Click here for the proof .
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INTEGRALBOT BETA-DIGAMMA INTEGRAL
Today we will show the following result that appears in this twitter post We will start by proving two lemmas involving the Beta and Digamma functions. Lemma 1: (1) Recall the Beta function (2) If we differentiate (2) w.r. to s we obtain Lemma 2: (3) Recall the Beta function (4) If we differentiate (4) w.r. to a we obtain Now let´s evaluate the integral: For the proof of the special values of the Digamma function see this post . Plugging the values of J and K back in the original integral we obtain:
\int_0^\infty \left(\frac{\sinh(ax)}{\sinh(x)}-\frac{a}{e^{2x}}\right)\,\frac{dx}{x}
In this post We will compute the following integral: Then We used that (see here ): and ( here )
Clausen Function, Logsine Integrals and Central Binomial Series
In today´s post we will review the Clausen function and prove some of its properties. Than we will show it´s connection with logsine integrals and Binomial series. The Clausen function is defined as Recall the Fourier expansion (see here ) (1) Integrating both sides from 0 to x we obtain (2) Claim 2: (3) Proof: On the other hand Hence Sometimes it´s useful to integrate the Clausen function: (4) Proof: (5) Proof: From equation (2) we can derive a duplication formula for (6) Proof: the same procedure we can obtain a duplication formula for (7) Proof: Integrating both sides of (6) form 0 to x we have: Lets evaluate each of these integrals separately Putting all together The general formula is given by: (8) First note that (very easy to prove just by expanding the R.H.S.): (9) Then (10) On the other hand (11) By (9) we can equate (10) and (11) (12) Proving the duplication formula for od...