Motivation for decomposition into even and odd functions

Every function can be expressed uniquely as the sum of an even function and an odd function, like so: let E(x) be (f(x)+f(-x))/2, and let O(x) be (f(x)-f(-x))/2. Then E(x) is even, O(x) is odd, and f(x) = E(x) + O(x).

But what’s the point of doing this?

Decomposing any function $f$ into its even and odd parts

$$ E(x) = \frac{f(x)+f(-x)}2,\quad O(x)=\frac{f(x)-f(-x)}2,\quad f=E+O $$

is more than just a neat algebraic trick. Here’s why it’s so useful:

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1. Exploiting symmetry in integrals

• Odd integrals vanish: if $O$ is odd, then

  $$ \int_{-a}^a O(x)\,dx = 0. $$

  So

  $$ \int_{-a}^a f(x)\,dx = \int_{-a}^a E(x)\,dx + \underbrace{\int_{-a}^a O(x)\,dx}_{0} = 2\int_0^a E(x)\,dx. $$

  You’ve halved your work by reducing to the even part only.

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2. Fourier series simplification

• An arbitrary periodic $f$ has a Fourier series

  $$ f(x)=a_0 + \sum_{n=1}^\infty\bigl[a_n\cos(nx)+b_n\sin(nx)\bigr]. $$

• The cosine terms are the even part; the sine terms are the odd part.

• You can immediately tell which coefficients vanish if you know $f$ is purely even or purely odd.

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3. Power-series (Taylor) clarity

• Expanding about zero, an even function has only even powers:

  $$ E(x)=c_0 + c_2 x^2 + c_4 x^4 + \cdots, $$

  while an odd function has only odd powers:

  $$ O(x)=c_1 x + c_3 x^3 + \cdots. $$

• Decomposition helps identify which terms must be present or zero in your series.

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4. Solving differential equations

• Many differential operators commute with the “flip” $x\mapsto -x$, so they preserve evenness/oddness.
• You can split a boundary-value or initial-value problem into two simpler problems—one on the even subspace, one on the odd—often halving the complexity.

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5. Linear-algebra viewpoint

• The space of all functions (say, on $\mathbb{R}$) decomposes as

  $$ \{\,\text{even}\}\ \oplus\ \{\text{odd}\}. $$

• Projection onto each subspace is exactly $E$ and $O$. This is a concrete example of projecting onto complementary subspaces in any vector space.

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6. Practical computations & modeling

• In physics and engineering, boundary conditions often enforce symmetry. If you know a solution must be even (or odd), you can discard half the domain or simplify the ansatz immediately.
• In signal processing, even/odd decomposition relates to splitting a signal into its symmetric and antisymmetric components, each of which may carry different information.

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In short

Breaking $f$ into $E+O$ lets you

1. Harness symmetry to simplify integrals and equations.
2. Organize expansions (Fourier, Taylor) by parity.
3. Project cleanly onto invariant subspaces.

It’s a fundamental tool whenever symmetry—or the lack of it—can be used to your advantage.