Fourier series for exponentials even or odd function?

[physics4life]

hi peeps. just a quick one.
(a) how would you go around working out the Fourier for exponential functions..
simply something like e^x? (b) and how can this be applied to work out Fourier series for cosh and sinh (considering cosh = e^x + e^-x / 2) etc etc..

first of all.. is e^x even or odd function..
i appreciate even function is: f(x) = f(-x)
odd function is : -f(x) = f(-x)

if for example , x =1.. e^x = e1...
so f(x) = e1
so e1 = 2.718...
e(-1) = 0.367... which is neither f(x) or -f(x)?? so there's a sticky point as its not clear whether this is even or odd..??

 

[d_leet]

Not all functions are even, or odd. Some are neither, f(x)=e^x is such a function.

 

[HallsofIvy]

e^x is neither odd nor even. Given any function, f(x), we can define the even and odd parts of f by
$$f_e(x)= \frac{f(x)+ f(-x)}{2}$$ and
$$f_o(x)= \frac{f(x)- f(-x)}{2}$$
In particular, the even and odd parts of e^x are
$$\frac{e^x+ e^{-x}}{2}= cosh(x)$$ and
$$\frac{e^x- e^{-x}}{2}= sinh(x)$$

 

[rayman123]

e^x is neither odd nor even. Given any function, f(x), we can define the even and odd parts of f by
$$f_e(x)= \frac{f(x)+ f(-x)}{2}$$ and
$$f_o(x)= \frac{f(x)- f(-x)}{2}$$
In particular, the even and odd parts of e^x are
$$\frac{e^x+ e^{-x}}{2}= cosh(x)$$ and
$$\frac{e^x- e^{-x}}{2}= sinh(x)$$

okej but what about such function then
$$f(x)=x^2e^{-x}$$ what kind of function do we get if we multiply an even function with a function that is neither odd nor even?

 

[rayman123]

ah I know
it is neither odd nor even