Interpolating an infinite sequence ?

[tommy1729]

\[
f(z) = A(z)\sin(2 \pi z) \sum_{n=0}^\infty \frac{a_n}{2\pi A(n) (z-n)}\\
\]

This satisfies:

\[
f(n) = a_n\\
\]

And you can force convergence of the series by letting  \(A(n)\) be as large as possible. So for example \(A(z) = e^z\) works.

How does f(5) = a_5 follow from 

\[
f(z) = \Exp(z)\sin(2 \pi z) \sum_{n=0}^\infty \frac{a_n}{2\pi \Exp(n) (z-n)}\\
\]

or 

\[
f(5) = \Exp(5)\sin(2 \pi 5) \sum_{n=0}^\infty \frac{a_n}{2\pi \Exp(n) (z-n)}\\
\]

??

regards

tommy1729

not sure why tex fails 

slightly better

How does f(5) = a_5 follow from 

\[
f(z) = Exp(z)\sin(2 \pi z) \sum_{n=0}^\infty \frac{a_n}{2\pi Exp(n) (z-n)}\\
\]

or 

\[
f(5) = Exp(5)\sin(2 \pi 5) \sum_{n=0}^\infty \frac{a_n}{2\pi Exp(n) (z-n)}\\
\]

??

regards

tommy1729