Let denote the trigamma function and let be integrable on . It holds that

**Solution**

We have successively:

where the interchange between the infinite sum and the integration is allowed by the uniform bound

The result follows.

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Let denote the trigamma function and let be integrable on . It holds that

**Solution**

We have successively:

where the interchange between the infinite sum and the integration is allowed by the uniform bound

The result follows.

Let denote the trigamma function , the digamma function and let be integrable on . It holds that

Proof:We have successively , using the aboveApplication: