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Tag Archives: Real Analysis
Fractional integral
Let denote the fractional part. Prove that
for the different values of the integer number .
Solution
Let denote the integral,
since if
whereas
if
. Therefore,
except of a countable set whose measure is
.
Multiple logarithmic integral
Let denote the Riemann zeta function. Evaluate the integral
Solution
Based on symmetries,
Let . It follows that
Using the recursion we get that
Thus,
An integral equality
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.
Digamma series
Let denote the digamma series. Evaluate the series
Solution
The series telescopes;
Continuous and periodic
Let be a continuous real-valued function on
satisfying
Define a function on
by
- Prove that
is continuous and periodic with period
.
- Prove that if
is continuous and periodic with period
then
Solution
- We note that
- First of all
is bounded on
and
uniformly. Hence,