# To find an entire function

DanniHuang

## Homework Statement

To find entire functions which satisfy g($\frac{1}{n}$) = g(-$\frac{1}{n}$) = $\frac{1}{n^{2}}$

## Homework Equations

How many functions can be found?

## The Attempt at a Solution

Because the function is entire, it can be expanded in the Taylor series. But how can I work out the question?

Homework Helper

## Homework Statement

To find entire functions which satisfy g($\frac{1}{n}$) = g(-$\frac{1}{n}$) = $\frac{1}{n^{2}}$

## Homework Equations

How many functions can be found?

## The Attempt at a Solution

Because the function is entire, it can be expanded in the Taylor series. But how can I work out the question?

Hint :

This condition here : g($\frac{1}{n}$) = g(-$\frac{1}{n}$)

Should look oddly familiar to : $g(x) = g(-x)$ which is the definition of an EVEN function.

For example, consider these functions :

f(x) = x^2
f(x) = x^4
...
f(x) = x^2n

;)

Homework Helper
You should be able to easily find one entire function that satisfies that. A hint might be, 'don't think too hard'.

DanniHuang
Hint :

This condition here : g($\frac{1}{n}$) = g(-$\frac{1}{n}$)

Should look oddly familiar to : $g(x) = g(-x)$ which is the definition of an EVEN function.

For example, consider these functions :

f(x) = x^2
f(x) = x^4
...
f(x) = x^2n

;)

So n can only be even numbers with the Ʃa$_{n}$z$^{n}$=$\frac{1}{n^{2}}$. And then?

Homework Helper
So n can only be even numbers with the Ʃa$_{n}$z$^{n}$=$\frac{1}{n^{2}}$. And then?

Not necessarily, consider : cos(x), cosh(x), |x|

Those are all even functions as well.

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