# Questions tagged [even-and-odd-functions]

Even functions have reflective symmetry across the $y$-axis; odd functions have rotational symmetry about the origin.

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### odd function conclusion in D'Alembert's formula

He there, here I have a particular question of D' Alembert's formula for the homogeneous wave equation in 1D: \begin{align}\frac{\partial^2 u}{\partial t^2} - C^{2}.\frac{\partial^2 u}{\partial x^2} =...
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### is 0 even if even numbers are only defined in for natural numbers? [duplicate]

I have a question, if even numbers are only defined in natural numbers how 0 is even? I understand that even is the number divisible by 2 so 0 is even but it is defined this way only for natural ...
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### How to determine if -h(-x) is even odd or neither

I know that h(x) is even so it definitely can't be A, so it could be one of the other 3 choices, but I am not sure how to determine if -h(-x) is even odd or neither.
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### Doubt about the power of $x$ of the function to find that $f(x) = 3$ is an even function

Using the definition of even functions, $f(-x) = f(x) = 3$, hence it is an even function. Also, it is being mentioned that if $f(x)$ is an even power of $x$, then it is an even function of $x$. Same ...
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### Parity in the Schrödinger equation of the hydrogen atom

Hi I'm studying quantum mechanics with Stephen Gasiorowicz's book. It says for the $n=2$ states of the hydrogen atom, the $l=0$ state has even parity and the $l=1$ state has odd parity. As far as I ...
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### Why is $e^{-ix}$ an odd function?

My physics prof pulled this out of the air to swap the limits on an integral: $\int_{0}^{\infty} e^{-ix} \, dx = \int_{-\infty}^{0} e^{ix} \, dx$ "because the integrand is odd." So far as I ...
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I am an adult learner trying to learn calculus again. I have learnt that for an odd periodic function with period, say, T: $$f\left(-t\right)=-f\left(t\right)$$ $$\int_{-\frac{T}{2}}^{\frac{T}{2}}{... 0 votes 1 answer 50 views ### let f be analytic in Ball_2\left(0\right) and f is odd. let U=\left\{z\in \mathbb{C}|1<\left|z\right|<2\right\}, prove equality Sorry for title being not full, could not write it all: let f be analytic in Ball_2\left(0\right) and f is odd. let U=\left\{z\in \mathbb{C}|1<\left|z\right|<2\right\} Prove: \exists f\in ... 2 votes 0 answers 71 views ### How to find constant a and real function f such that: \Im f(a+ix)=\ln(x) I have a question. Given a real function g(x), is it possible to find some constant a and a real function f such that$$\Im f(a+ix)=g(x)$$For example,$$f(x)=e^x,~~~~\Im f(ix)=\Im e^{ix}=\sin x=... 1 vote
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### Proving that the odd coefficients of the polynomial that interpolates an even function are zero

Let $n \in \mathbb{N}$ be fix, $a>0$ and $f:[-a,a] \rightarrow \mathbb{R}$ be an even funciton. Consider the polynomial $p(x) = \sum\limits_{i =0}^n c_ix^i \in \mathbb{P}_n$ that interpolates $f$ ...
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### Integrals, and even and odd functions

I am looking at a proof and got stuck on a part with an integral. I tried to simplify the problem as much as possible, I hope I did not omit any potential helpful information. I have an even ...
### Is $\cos(a(\pi-t))$ an even function?
I've an assignment due in a couple of days, and I'm wondering if my teacher made a mistake in the question below, or if I'm missing something silly. Let $a \in [0, 1]$ and $f_a$ the $2\pi$-periodic ...