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I know how to find Fourier series of a function,but i found following question and I stuck.

"Show that any function $f(x)$ defined on symmetrically placed intervals can be written as sum of an even function and an odd function .

Hence show that how to write $f(x) =x+x^2+x^3$ as per above statement.

Can anyone please help me...thanks in advance.

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  • $\begingroup$ Do not use Fourier series. Given any $f(x)$, find $g(x)$ and $h(x)$ in terms of $f(x)$ so that $f(x) = g(x) + h(x)$. For example $g(x) = e^{f(x)} + e^{f(-x)}$ is even, but of course you can make a much simpler function $g$ that is even. $\endgroup$
    – Michael
    Commented Oct 21, 2019 at 16:09
  • $\begingroup$ Can you tell which terms of your $f(x)$ are even and which are odd? Group them accordingly. $\endgroup$
    – MPW
    Commented Oct 21, 2019 at 16:10
  • $\begingroup$ @MPW x and x^3 are odd and x^2 is even $\endgroup$
    – Cent22
    Commented Oct 21, 2019 at 16:12
  • $\begingroup$ So doesn't that mean that $f(x)=\underbrace{(x^2)}_{\textrm{even}} + \underbrace{(x+x^3)}_{\textrm{odd}}$ fits the bill? $\endgroup$
    – MPW
    Commented Oct 21, 2019 at 16:14
  • $\begingroup$ @MPW now i got it...f(x)=h(x)+g(x),. Where h(x)=x^2 is even and g(x)=x+x^3 is odd....thanks $\endgroup$
    – Cent22
    Commented Oct 21, 2019 at 16:24

1 Answer 1

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Suppose $f(x)=h(x)+g(x)$ where $h(x)$ is even, $g(x)$ is odd. Then $f(-x)=h(x)-g(x)$. Adding both equations we get that $h(x)=\frac{f(x)+f(-x)}{2}$ and $g(x)=...$

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    $\begingroup$ g(x)=f(x) - f(-x)/2...thank you sir $\endgroup$
    – Cent22
    Commented Oct 21, 2019 at 16:27

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