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At first, we sample in the ( is odd) equidistant points around :

where is some step.

Then we interpolate points by polynomial

(1)

Its coefficients are found as a solution of system of linear equations:

(2)

Here are references to existing equations: (1), (2).

Here is reference to non-existing equation (??).

Compilation of an expression may be suppressed, showing instead the LaTeX source, by preceding the expression with a !.

For mathematical graphs you may use tikzpicture and pgfplots, e.g. :

Binomial theorem

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Exponential function

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Cauchy–Schwarz inequality

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Bayes’ theorem

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Euler’s summation formula

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_Theorem._ Euler’s summation formula. _If has a continuous

derivative on the interval , where ,

then_

(3)

_Proof._ Let , . For integers and

in we have

Summing from to we find

(4)

Integration by parts gives us

When this is combined with \eqref{summation} we obtain

\eqref{theorem}.

## 3 Responses

Let be a positive integer and be the fractional par function, then calculate in closed-form the following Multiple Integral :

At first, we sample in the ( is odd) equidistant points around :

where is some step.

Then we interpolate points by polynomial

(1)

Its coefficients are found as a solution of system of linear equations:

(2)

Here are references to existing equations: (1), (2).

Here is reference to non-existing equation (??).