Decompose Even & Odd parts of a function?

[radiator]

How would you decompose a given function to its even and odd parts? let's say you have f(x)=e^ix, and would like to know the even and odd parts of it? how do you proceed?

Thank you

 

[arildno]

It is not particularly meaningful to decompose complex-valued functions as "even" or "odd".

however, any function f(x), can be written as the sum of two functions:
f(x)=(f(x)+f(-x))/2+(f(x)-f(-x))/2

 

[radiator]

Thanks arildno,
I suppose if I have a step function, the even part will be equal to odd part?
i.e. f(x) = (H(x) + 0)/2 + (H(x) - 0 )/2
since H(x) = 0 for x<0

 

[arildno]

Not really.

For positive x, H_even(x)=1/2, and of course, for negative x, we have H_even(x)=1/2

The odd part of the Heavyside function is, for x>0, H_odd(x)=1/2, whereas for x<0, H_odd(x)=-1/2.

 

[radiator]

I get it, but just to further understand more

suppose again I have f(x) = x^2 , which means the f_e(x) = x^2 and f_o(x) = 0, similarly f(x) = x^3 gives f_e(x) = 0 and f_o(x) = x^3 which is graphically a reflection around the y-axis and the origin, respectively.

but for absolute x it would be f_e(x) = (x+x)/2 = x for x>0 and f_e(x) = (x+(-(-x)))/2 = x
f_o = (x-x)/2 = 0 for x>0 and f_o(x) = (x-(-(-x)))/2 = 0 for x<0 , is this right ?

btw I though latex works here but I can't find the correct code, test: <latex> $\beta$ <\latex>

 

[arildno]

Now, for x>0, we have:

|x|=(|x|+|-x|)/2+(|x|-|-x|)/2, that is =(x+-(-x))/2+(x--(-x))/2=x+0, as it ought to be

For x<0, we have:
|x|=(|x|+|-x|)/2+(|x|-|-x|)/2=(-x+-x)/2+(-x--x)/2=-x+0, as it ought to be.

Thus, f_(e)(x) equals the absolute value function itself. Also as it ought to be, I may add.

 

[radiator]

Thanks very much arildno :)