Real Function is Expressible as Sum of Even Function and Odd Function
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Theorem
Let $f: \R \to \R$ be a real function which is neither an even function nor an odd function.
Then $f$ may be expressed as the pointwise sum of an even function and an odd function.
Proof
Let:
\(\ds \map g x\) | \(=\) | \(\ds \dfrac {\map f x + \map f {-x} } 2\) | ||||||||||||
\(\ds \map h x\) | \(=\) | \(\ds \dfrac {\map f x - \map f {-x} } 2\) |
We note that:
\(\ds \map g {-x}\) | \(=\) | \(\ds \dfrac {\map f {-x} + \map f {-\paren {-x} } } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\map f {-x} + \map f x} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map g x\) |
Thus $g$ is an even function.
Then:
\(\ds \map h {-x}\) | \(=\) | \(\ds \dfrac {\map f {-x} - \map f {-\paren {-x} } } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\map f {-x} - \map f x} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac {\map f x - \map f {-x} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\map h x\) |
Thus $h$ is an odd function.
Then:
\(\ds \map g x + \map h x\) | \(=\) | \(\ds \dfrac {\map f x + \map f {-x} } 2 + \dfrac {\map f x - \map f {-x} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\map f x } 2 + \dfrac {\map f {-x} } 2 + \dfrac {\map f x } 2 - \dfrac {\map f {-x} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map f x\) |
Hence the result.
$\blacksquare$
Examples
Arbitrary Function
Let $f: \R \to \R$ denote the real function:
- $\map f x = e^{2 x} \sin x$
$f$ can be expressed as the pointwise sum of:
- the even function $\map g x = \dfrac {\paren {e^{2 x} - e^{-2 x} } \sin x} 2$
and:
- the odd function $\map h x = \dfrac {\paren {e^{2 x} + e^{-2 x} } \sin x} 2$
Sources
- 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.3$ Functions of a Real Variable: $\text {(h)}$ Even and Odd Functions $(9)$