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Question
given function f(x)on (-L,L)
If f(x) is even(cosine series) ,odd(sine series) ,either(Fourier series)
what f is given on arbitrary interval(a,b) ????
I don't believe a function on the interval (a,b) is referred to as even or odd, but only when the range is (-L,L). This is because for every point at which the function exists, say x,
it also needs to exist at -x.
This is because of the last two rules of such functions. They are (1) The integral of an odd function from −A to +A is zero (where A is finite, and the function has no vertical asymptotes between −A and A) and (2) The integral of an even function from −A to +A is twice the integral from 0 to +A (where A is finite, and the function has no vertical asymptotes between −A and A).
This means that if the function is not defined at both -A and A, it is not even or odd.
I found a place that has the definition of even and odd functions.
It is http://en.wikipedia.org/wiki/Even_and_odd_functions and that is where I got the preceeding rules from.
Here, the functions are defined on all real numbers.
Here are some other rules out of that document:
1) The only function which is both even and odd is the constant function which is identically zero (i.e., f(x) = 0 for all x).
2) Ths sum of an even and odd function is neither even nor odd, unless one of the functions is identically zero.
3) Even+Even=Even, Odd+Odd=Odd
The sum of two even functions is even, and any constant multiple of an even function is even.
The sum of two odd functions is odd, and any constant multiple of an odd function is odd.
4) Even*Even = Odd*Odd = Even; Even*Odd=Odd
The product of two even functions is an even function.
The product of two odd functions is an even function.
The product of an even function and an odd function is an odd function.
5) Quotients
The quotient of two even functions is an even function.
The quotient of two odd functions is an even function.
The quotient of an even function and an odd function is an odd function.
6) Derivatives, where they go odd-> even and even-> odd
The derivative of an even function is odd.
The derivative of an odd function is even.
7) Composition
The composition of two even functions is even, and the composition of two odd functions is odd.
The composition of an even function and an odd function is even.
The composition of any function with an even function is even (but not vice versa).
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