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So I learned about this formula $$ f(x) = \frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2} $$ and I'm wondering, is it of any value to express a function in this form?

kptlronyttcna
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    It's definitely as useful as expressing a matrix as the sum of a symmetric and a skew symmetric matrix, if not more. – Nick Sep 03 '14 at 15:49
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    Suppose I want to compute the integral $\int_{-a}^a f(x),dx$. If I split $f(x)$ up as above, then by symmetry 1) the integral over the odd part above vanishes, 2) the remaining integral over the even part may be written as $\int_0^a (f(x)+f(-x)),dx$. So if that sum is simpler than $f(x)$ itself, then I've made it easier to do the indefinite integral – Semiclassical Sep 03 '14 at 15:50

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There's a further value beyond the ones mentioned in the comments, which is that this splitting up of a function -- along with decomposing a matrix as a sum of a symmetric and skew-symmetric one, and expressing a complex number as one that's invariant under conjugation (a real) and one that negates under conjugation (a pure imaginary), and expressing a vector as the sum of its perpendicular projection onto a subspace $W$ and its projection onto the orthogonal subspace $W^\perp$ -- is a technique that gets used over and over: to find the "even" part, you average the function with its reflection, and the result HAS to be even. The same technique works to do all those other things (you average the matrix with its transpose, etc.).

There are two ways to look at this.

One is that if you have a transformation (like "negating the argument of a function") which, when done twice, amounts to the identity, and the transformation is linear, then the domain of the transformation splits as a direct sum of the $+1$ and $-1$ eigenspaces. If you have not studied linear algebra, that's just gobbledegook to you, and I apologize -- the point is that there's a general framework in which these various tricks can be understood as part of a coherent whole.

Another way is to say that "negating the argument" is an operation applied to functions, as is "keeping the argument the same". These two operations form an algebraixc structure called a "group", and what you have is a "group action" of this two-element group on the set of functions from $R$ to $R$. The critical step in the splitting is to take a function and 'act on it' by every element of the group (there are only two) and average the result. This trick' --- averaging over a group action --- is another one that keeps producing interesting results in math, so it's nice to know about (and to see examples early!). One example is that this is how the Haar measure for toplogical groups is created. (Again, I apologize if this is beyond your current mathematical experience.)

John Hughes
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