Questions tagged [even-and-odd-functions]

Question 1

Given a real and odd signal $x(t)$, such that $\vert X(\omega)\vert = e^{-\vert \omega\vert}$ is the magnitude of Fourier transform. Question: Find the Fourier Transform $X(w)$. My attempt: We know, $...

Question 2

Into this textbook Book of Integrals: Exploring Species of Integrals and Their Techniques of Miguel Santiago, I have read The S.E-One Method. Supposing to evaluate a definite integral over symmetric ...

Question 3

Consider functions $f$ which are involutions, i.e. \begin{align} f(f(x))=x\quad \implies \quad f'(x)f'(f(x))=1. \end{align} Under the (Legendre-like) contact transformation \begin{align} f(x)=F'(X),\ ...

Question 4

The following is an algebra exercise for public university applicants, where an increasing (or decreasing) function is understood in the strict sense: Determine if the following statements are True ...

Question 5

I am tackling some minor Analysis topic, which has nonetheless to do with rational functions and how they are integrated. My textbook says the following: If a rational function $R(u, v)$ is unaltered ...

Question 6

Recently, it has come to my attention that $f(x)=\lfloor x \rfloor +\frac{1}{2}$ is an odd function. Verification: For $x\notin\Bbb{Z},$ using $\lfloor x \rfloor + \lfloor -x \rfloor=-1$ we get that $$...

Question 7

I am trying to interpret and visualize the effect that certain boundary conditions have on the future evolution of a wave. Assume $f(t,x)$ is solution to some wave equation in a box $x \in [-1,1]$ ...

Question 8

In A Transition to Advanced Mathematics 8e pg. 52, there are two example proofs back to back that seem to contradict each other. One example proves "The function $f$ given by $f(x) = x^3 - \...

Question 9

$α,β,γ\in\Bbb R,$ $$f(α,β,γ):=\sin (α-β-γ) \sin (α+β-γ) \sin (α-β+γ) \sin (α+β+γ)$$ Clearly, the maximum is $f(\frac{π}{2},\frac{π}{2},\frac{π}{2})=1$. I guessed the minimum is $-1$, but$$f(α,β,γ)=f(-...

Question 10

My question comes from the book Stable Solutions of Elliptic Partial Differential Equations Louis Dupaigne, pages 30-32. Summary: Which uniqueness theorem to use for this differential equation ? I am ...

Question 11

Consider the following matrix $$M := \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 &...

Question 12

I used Mathematica to calculate the antiderivative of $\cos (\pi x)/x$. I obtained the cosine integral $$ \int \frac {\cos (\pi x)}{x} dx = Ci(x) $$ where $$ \begin{aligned} Ci(x) &:= - \int_x^\...

Question 13

Let $\varphi: \mathbb{R} \to \mathbb{R}$ be a periodic function of period $L>0$, that is, \begin{equation}\label{periodicitycondition} \varphi(x+L)=\varphi(x),\; \forall\; x \in \mathbb{R}. \tag{1} ...

Question 14

As the definition goes, a function $f(x)$ is even if $f(-x)=f(x)$ and it is odd if $f(-x)=-f(x)$, in which the domain is not paid enough attention to. For example, $f(x)=x^2$ is even for any symmetric ...

Question 15

Let $n \in \mathbb N$. Let's say a complex function $f: U \rightarrow \mathbb C$ is "of type $k \pmod n$" if for one (and hence every) primitive $n$-th root of unity $\omega$, $$f(\omega z) =...

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Questions tagged [even-and-odd-functions]

Computing Fourier transform for a real odd signal

The S.E-One Method for a particular class of functions

Every odd function has a corresponding involution

There exists at least one even function that is increasing

Why can a rational function even in some argument be represented as a function of the square of that argument?

Odd functions like $\lfloor x \rfloor + 1/2$

Interpretation of boundary conditions

How can the statements "$f(x) =x^3 - \frac4x$ is an odd function" and "if $f$ is an odd function, then $f(0)=0$" both be true?

Find the minimum of $\sin (α-β-γ) \sin (α+β-γ) \sin (α-β+γ) \sin (α+β+γ)$

How to prove that solutions of semilinear differential equations is even function?

In what sense is this column-sum=1, row-sum=2 matrix even, since the imaginary part of its Fourier vanishes?

Puzzled by asymmetry of cosine integral

Translation of odd and even functions

Even/Odd with reference to the interval (domain)

Anything interesting known about this generalization of even and odd functions?

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