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What is the real world use of knowing whether a function is odd or even? Any practical examples?

For example, quadratic equations, differential equations and calculus in General is used for, among other things, determining motion of bodies. Just knowing that a function is symmetrical or not, help how? In terms of solving practical problems with the knowledge.

partlycloudy
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7 Answers7

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For a more physics-y answer, you can turn to quantum mechanics. For example,

  • If a wavefunction of a particle is an even function (symmetric, centered at the origin), then the expected location of the particle (assuming it exists) is the origin, because $|\psi(x)|^2$ is even, so $x |\psi(x)|^2$ is odd, so $\mathbb E (x) \equiv \int x|\psi(x)| \mathrm d x=0$. More generally, the odd moments of an even probability distribution vanish.
  • A wavefunction has definite parity if it is odd or even. (It is an eigenvector with eigenvalue $\pm 1$ of the parity operator $\mathrm P$.) One can show that generically, for a symmetric (even) potential, the energy eigenstates have parity even, odd, even, odd, ... This leads to simple numerical strategies for trying to calculate the lowest two eigenstates by restricting to even (for ground state) and odd (for lowest excited state) functions.

There are plenty of other generic properties, like

  • Odd derivatives of even functions and even derivatives of odd functions vanish, so ...
  • ... analytic even functions have power series containing only even powers of $x$...
  • ... and in fact also Fourier series containing only $\cos$ terms.
not all wrong
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An area of mathematics which is often used in engineering is Fourier series. There you rewrite a periodic function as an infinite sum and you have a formula for the coefficients. If a function is even or odd, half of the coefficients are zero.

Michael Albanese
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The Borsuk-Ulam theorem has lots of equivalent formulations in terms of odd and even functions. For example:

A continuous odd function $f \colon S^n \to \Bbb{R}^n$ vanishes at some point.

An application of this theorem, that seems to be something about the real world is the following (this might not strictly be true, depending on your assumptions):

At any point of time, there are two antipodal points on the earth, with the same temperature and pressure.

For more such applications, I suggest you look at a book like "Using the Borsuk–Ulam theorem" by Jiří Matoušek.

Community
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ronno
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For any $a\in\mathbb{R}$, $$\int_{-a}^a \sin x dx=0$$

vadim123
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It saves computation time to only draw or calculate half of something, especially when you consider the fact that if f(x) and g(x) are even, f(g(-x))=f(g(x)) and if they're odd then f(g(-x))=f(-g(x))=-f(g(x)), which means the thing you're calculating could be an approximation of an infinite composition of even or odd functions and you'd still be able to exploit this symmetry.

If f and g are even, f(x)+g(x)=f(-x)+g(-x), f(-x)g(-x)=f(x)g(x), -f(x)=-f(-x), and 0 is even, so the even functions form a ring without unit. Studying rings brings overall happiness and fulfillment, which leads to a longer life.

Loki Clock
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A real-world obvious example of an even function is the deflection of a beam supported at two points with an equal distribution of forces at both sides or the displacement of a thin wire fixed at both sides horizontally when you pull it at the middle. In both examples it is supposed that the origin is at the center of the physical system.

Mostafa Jani
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The fact of a function being odd or even tells you about symmetry, which is always useful.

Constantine
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