# Bernoulli Polynomials

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Bernoulli Polynomials are defined and then various properties are demonstrated.

Key words:

1) Contour Integral Definition

2) Euler Numbers

3) Fourier Series

4) Evaluation of Riemann Zeta function for even integers

5) Stirling Numbers of the Second Kind

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Bernoulli Polynomials are defined and then various properties are demonstrated. The solution is detailed and well presented.

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Bernoulli Polynomials

1) The Bernoulli Polynomials can be defined as:

for n>0

2) The Bernoulli Numbers are defined as

3) Consider the generating function for these polynomials:

To derive an expression for G(x,t) consider

Hence for some function A(t)

Now consider

But also

Conclude

Now the generating function is exceptionally powerful and will be used to derive all the other properties of the Bernoulli Polynomials found ...

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