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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In what sense is this column-sum=1, row-sum=2 matrix even, since the imaginary part of its Fourier vanishes?
Consider the following matrix
$$M := \begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 &...
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Puzzled by asymmetry of cosine integral
I used Mathematica to calculate the antiderivative of $\cos (\pi x)/x$. I obtained the cosine integral
$$
\int \frac {\cos (\pi x)}{x} dx = Ci(x)
$$
where
$$
\begin{aligned}
Ci(x) &:= - \int_x^\...
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Reference Request: Definition on parity of scalar-valued multivariate functions (even or odd)
I am studying the densities of multivariate Gaussian distributions and I am curious about the parity of those scalar-valued multivariate functions.
For a function $f(\mathbf{x}): \mathbb{R}^n \...
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Translation of odd and even functions
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}
...
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Even/Odd with reference to the interval (domain)
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 ...
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Anything interesting known about this generalization of even and odd functions?
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 $\omega$,
$$f(\omega z) =...
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Function that grows asymptotically as $x\log(x)$ or $\log(x!)$, but is even? Function that grows as $\log(x)$ but is odd?
I'm trying to implement a model for predicting origin location - destination location traffic, from road traffic (if you are interested in the model, it is from this paper: Bell1983). The author ...
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Possibilities for even quadratic
Is the only possibility for an even quadratic $ax^2$ + b where a and b are constants? Also is it necessary for the coefficients to be real?
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When does a sum of an odd and even function gives us either an odd or an even function?
Greetings,
I'm currently trying to find a way to prove that this functional equation has no solution:
$$f(x+yf(x))+f(xf(y)−y)=f(x)−f(y)+2xy^2$$
I know that an eventual solution has to be odd: to do so,...
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Question regarding proof whether a function is many-one
This question is in regards to the following problem
If
$$
f(x) = \bigl(h_1(x) - h_1(-x)\bigr) \cdot \bigl(h_2(x) - h_2(-x)\bigr) \cdots \bigl(h_{2n+1}(x) - h_{2n+1}(-x)\bigr)
$$
and $f(200) = 0$, the ...
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Even function given a condition
I would like to understand the rationale behind the solutions I found for the problem below.
Show that every function $f: \mathbb{R}^* \rightarrow \mathbb{R}$, satisfying the condition $f(xy)=f(x)+f(y)...
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generating function, compositions odd, congruence modulo
Find the generating function for the number of compositions of $n$ with an odd number of parts, each of which is congruent to $1 \bmod 3$.
We have $k$ parts where the total of the $k$ parts must be ...
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Proving the derivative of an even function is odd using the chain rule
My answer:
Suppose that f: ℝ $\rightarrow$ ℝ is an even function, that is differentiable everywhere. If f is an even function, then we have that f(x)= f(-x). We now take the derivative using the chain ...
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Parity of solutions to the Abel equation of the second kind $yp-y=f(x)$ for even and odd $f(x)$
Consider the Abel equation of the second kind,
\begin{align}
yp-y=f(x);\quad [p=y'_x]
\end{align}
for some arbitray non-zero function $f$.
Suppose $y$ were an odd function, then $p$ is even and it ...
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Let $F(x)= \int_{0}^{x} f(t) \, dt$. Show that if f is even, F is odd. [duplicate]
I have been working on this question. I know that if a function is even then f(x)=f(-x). Take derivative of both sides and you get f'(x)=-f'(-x), hence f' is odd. However, I am not sure how to go ...
user1172934
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Claims about solution of nonlinear differential equation with symmetries
I have a nonlinear ordinary differential equation (of sixth order). If $y(x)$ satisfies the given nonlinear differential equation and its associated boundary conditions, both $y(1-x)$ and $-y(1-x)$ ...
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For which polynomials $f$ does there exist a $g$ with $g\circ f$ even?
For which polynomials $f(x)$ does there exist a nonconstant polynomial $g(x)$ such that $g\big(f(x)\big)$ is an even function?
If $f$ is already even, then $g$ can be the identity. If $f$ is odd, then ...
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$f(f(x)) = x^3$, can $f$ be proved to be an odd function [duplicate]
Further if
$$f(f(x)) = g(x)$$
and we know that $g(-x)=-g(x)$ , i.e it's an odd function can we determine whether $f$ will be odd or not ?
Coming back to the original question, I first determined that $...
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best polynomial approximations to $f$ vanish at $0$ imply $f$ is an odd function
It is known that if $f\in C[-1, 1]$ is odd/even, then the best polynomial approximation (in $L^{\infty}$ norm) of degree $n$, denoted by $p_n$, must also be odd/even. This has been asked on MSE before....
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is 0 even if even numbers are only defined in for natural numbers? [duplicate]
I have a question, if even numbers are only defined in natural numbers how 0 is even? I understand that even is the number divisible by 2 so 0 is even but it is defined this way only for natural ...
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How to determine if -h(-x) is even odd or neither
I know that h(x) is even so it definitely can't be A, so it could be one of the other 3 choices, but I am not sure how to determine if -h(-x) is even odd or neither.
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The Odd/Even Extensions and Why We even do Them?
I have the following problem. Let
$$
f(x) =
\begin{cases}
x & 0\leq x \leq \frac{\pi}{2}, \\
x-\pi & \frac{\pi}{2} \leq x \leq \pi
\end{cases}
$$
and let $h$ be the odd extension of $f$ to $...
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The motivation for involving complex conjugation in the definition of odd/even complex-valued functions
As of November 19, 2023, the Wikipedia page for even and odd functions defines odd/even symmetric complex-valued functions as below:
even: $f(x)=\overline{f(-x)}$,
odd: $f(x)=-\overline{f(-x)}$.
What ...
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Even or odd, periodic or not $k>0 \; ; \forall x\in\mathbb{R}, f(x+k)=f(k-x)\; and\; f(2k+x)=-f(2k-x) $
I need to know if the function $f$ are odd or even, and I need to know if it periodic or not.
$$k>0 \; ; \forall x\in\mathbb{R}, f(x+k)=f(k-x)\; and\; f(2k+x)=-f(2k-x) $$
I've tried to substitute x ...
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How should I understand $\frac {dx(-t)} {d(-t)}$ and relate it to $\frac {dx(t)}{dt}$? This is relevant to ‘Time Reversal Symmetry Theory’.
Consider a simple classical system that may execute simple harmonic motion, with equation of motion
\begin{equation*}
\frac{ d^2q(t) } { dt^2 }= - \omega^2q(t) \tag{1}
\end{equation*}
I wish to prove, ...
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vote
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Solving questions about correlation with symmetry
I try to determine if $X$ and $Y$ are (un)correlated and (in)dependent, when $X = \sin(\theta)$, $Y = \cos(\theta)$ and $\theta \sim N(0,1)$.
I know about the following formula:
$corr(X,Y) = \frac{...
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Parabolic sine-Gordon equation with periodic and odd initial data function
Consider the following parabolic equation
$$
\begin{cases}
\partial_t u = \partial_{xx}u + \sin u,\ (x,t)\in\mathbb{R}\times(0,\infty), \\
u(x,0) = u_0(x),
\end{cases}
$$
in which $u_0(x):\mathbb{R}\...
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If $f(x)$ is an even function, and $f(g(x))$ is an even function, then $g(x)$ MUST be?
If $f(x)$ is an even function, and $f(g(x))$ is an even function, then $g(x)$ must be:
(a) odd
(b) even
(c) general
(d) even or odd
(e) even or general
(f) even or odd or general
This question is in ...
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Generalisation of the idea of decomposition into even and odd functions
We know that a function $f(z)$ can be decomposed into $\frac{f(z)+f(-z)}{2}$ and $\frac{f(z)-f(-z)}{2}$. These are called the even and odd components.
I have made a generalisaiton of this. Suppose $f(...
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Doubt about the power of $x$ of the function to find that $f(x) = 3$ is an even function
Using the definition of even functions, $f(-x) = f(x) = 3$, hence it is an even function.
Also, it is being mentioned that if $f(x)$ is an even power of $x$, then it is an even function of $x$. Same ...
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Parity in the Schrödinger equation of the hydrogen atom
Hi I'm studying quantum mechanics with Stephen Gasiorowicz's book.
It says for the $n=2$ states of the hydrogen atom, the $l=0$ state has even parity and the $l=1$ state has odd parity.
As far as I ...
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Why is $e^{-ix}$ an odd function?
My physics prof pulled this out of the air to swap the limits on an integral:
$\int_{0}^{\infty} e^{-ix} \, dx = \int_{-\infty}^{0} e^{ix} \, dx$
"because the integrand is odd."
So far as I ...
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Integrals of odd function proof [duplicate]
I am an adult learner trying to learn calculus again. I have learnt that for an odd periodic function with period, say, T:
$$
f\left(-t\right)=-f\left(t\right)
$$
$$
\int_{-\frac{T}{2}}^{\frac{T}{2}}{...
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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 ...
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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,
$$f(x)=e^x,~~~~\Im f(ix)=\Im e^{ix}=\sin x=...
user1194668
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answers
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Arranging numbers along lines
I have to arrange the numbers $1,\ 2,\ ...,\ 9$ on the points in the lines such that every two lines the sum of their elements is equal.
My Solution:
We already know that $1 + 2 + ... + 9 = 45$
Let $...
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Integrate without using the property that the function is odd
Integrate using substitution(not trigonometric) the following. I will share how I did it, even though the answer is not right.
$$\int_{-1}^1 x \sqrt{1-x^2} dx $$
$$ Substituting \space x^2=y $$
$$...
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Question on even, odd complex valued function
I was solving following multiple choice question (more than one options may be correct) on complex analysis.
Question: The function $f(z)=z^2$ where $0 \leq \arg(z) \leq \pi$ is not
(a) even
(b) odd
(...
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Why is the decomposition of a function into odd and even parts interesting?
For all functions $f:\mathbb{R}\to\mathbb{R}$ one can find a unique decomposition $f(x)=E(x)+O(x)$ where $E(-x)=E(x)$ and $O(-x)=-O(x)$.
Is there any branch of mathematics where analysing the ...
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Do functions over a ring have an odd and an even part?
All functions $f(x):\mathbb{R}\to\mathbb{R}$ can be decomposed into an even and an odd part $f(x)=E(x)+O(x)$.
The proof I see here, and on Wikipedia requires $2$ to have an inverse, however I want to ...
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Is $y^2$ an even function?
yesterday I asked this question
the solution of this problem says that since $y^2$ is an even function and so is $cosx-1$ therefore $y'$ is an even function which implies $y$ is odd function and from ...
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Is $y^2$ even function??
the solution of this problem says that since $y^2$ is an even function and so is $cosx-1$ therefore $y'$ is an even function which implies $y$ is odd function and from there on the solution builds up
...
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Error in the notes about even/odd function?
Exercise:
Be $f,g: \mathbb{R}\to\mathbb{R}$ two functions. $f$ is odd, and $g$ is even.
Prove that $f(g(x))$ is odd, and $g(f(x))$ is even.
I personally think there is a mistake since both the ...
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Integration of odd and even functions. Where is a flaw in my proof?
If the function $f$ has the property $f(a+x) = -f(a-x)$ $\forall x$, where a is a constant, then
$$\int_{(a-2)}^{(a+2)} f(x)dx = 0$$.
I have to deside whether this statement is true or false, and ...
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Is the set of natural numbers that are multiples of at least 3 numbers (including 1 and themselves) asymptotically even?
I was just thinking about this in my head, I am curious as to whether I am thinking about this correctly, or if there is a flaw in my thinking (especially about infinity)
My conjecture is that the set ...
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Why Even-Odd method for $\int_{-\pi/3}^{\pi/3} \ln(\sin(x))$ dx doesn't work
For $ \int_{-\pi/3}^{\pi/3} \ln(\sin(x)) $ dx, I tried using the Even-Odd method
Even part = $\frac{\ln(\sin(x)) + \ln(\sin(- x))}{2}$ = $\frac{\ln(\sin(x)) + \ln(-1) + \ln(\sin(x))}{2}$ = $\frac{\ln(-...
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How to justify double integral evaluating to 0 using symmetry of the multivariable function?
Given $f(x,y) = sin(y-2x+1)$ on the rectangular domain $R = [0, 1] \times [-1, 1]$, how is symmetry used to explain why the double integral of $f(x,y)$ over this domain is $0$?
Past attempts I used ...
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Odd initial condition makes heat equation odd
Is my proof correct for saying that if the diffusion equation has an odd initial condition then the diffusion equation must be odd.
I have this:
$$\partial_{t} u - \partial_{xx} u = 0\\
u(x,0) = f(x)\\...
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votes
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answer
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Determining when the composition of two functions is even or odd
A function is even $\iff$ $f(-x) = f(x)$ and odd $\iff f(-x) = -f(x)$.
If I have some function $f$ that is even and some function $g$ that is even, their composition is $f(g(x))$, right?
When I'm ...
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Determining whether a Function is Even or Odd given a specific Domain
I have learned that,to determine if a function is even or odd, I should use the following Formula:
if f(-x)=f(x) ---> f(x) is Even.
if f(-x)=-f(x)---> f(x) is Odd.
if neither, well then neither.
...
