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To find an entire function

  • Thread starter DanniHuang
  • Start date
  • #1

Homework Statement



To find entire functions which satisfy g([itex]\frac{1}{n}[/itex]) = g(-[itex]\frac{1}{n}[/itex]) = [itex]\frac{1}{n^{2}}[/itex]

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?
 

Answers and Replies

  • #2
Zondrina
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Homework Statement



To find entire functions which satisfy g([itex]\frac{1}{n}[/itex]) = g(-[itex]\frac{1}{n}[/itex]) = [itex]\frac{1}{n^{2}}[/itex]

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([itex]\frac{1}{n}[/itex]) = g(-[itex]\frac{1}{n}[/itex])

Should look oddly familiar to : [itex]g(x) = g(-x)[/itex] 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

;)
 
  • #3
Dick
Science Advisor
Homework Helper
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You should be able to easily find one entire function that satisfies that. A hint might be, 'don't think too hard'.
 
  • #4
Hint :

This condition here : g([itex]\frac{1}{n}[/itex]) = g(-[itex]\frac{1}{n}[/itex])

Should look oddly familiar to : [itex]g(x) = g(-x)[/itex] 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[itex]_{n}[/itex]z[itex]^{n}[/itex]=[itex]\frac{1}{n^{2}}[/itex]. And then?
 
  • #5
Zondrina
Homework Helper
2,065
136
So n can only be even numbers with the Ʃa[itex]_{n}[/itex]z[itex]^{n}[/itex]=[itex]\frac{1}{n^{2}}[/itex]. And then?
Not necessarily, consider : cos(x), cosh(x), |x|

Those are all even functions as well.
 
Last edited:

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