Question

@meta-math
show how any function can be written as a sum of odd and even functions

Answer 1

possibly all the functions which aren't odd or even.

Answer 2

let fx be any function,

then it can be expresed as

{f(x) +f(-x) }/2 + {f(x) - f-(x)}/2

and clearly one of them is odd and one is even,

cool isnt it!!!!!

Answer 3

so all, functions can be expressed as the sum of an odd and an even function.

surely this must be useful somehow..

where can i use this theorem?

Answer 4

one good use of it is to solve integration problems,
when we integrate from +x to -x,

then we can break the function,
the odd part gets cancelled , and we r left with even part only,

there is a particular class of problems based on theis concept,
(vvery limited though)

Answer 5

can i use this in Fourier transforms?

Answer 6

i dont know about that buddy!

Answer 7

i dont even know what fourier transformation is

Answer 8

sounds cool though

Answer 9

well you kinda assume your function is made of an infinite sum of sine and cosine functions

Answer 10

The result is even sharper: any function f : R --> R can be written as the sum of a unique odd function and a unique even function.

Because of this, yes @rhaukus, this is a way to simplify calculations of Fourier series. The odd part of the function corresponds to the sum of sin terms and the even part to the sum of cos terms.