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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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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Which of the following is an even function, regardless of the choice of f(x)? [closed]
For this question is there some particular way of solving this? I am trying to solve it but I have no idea where to start.
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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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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
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=...
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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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What is this function syntax and show does it show the given?
I’ve just begun learning composite functions and the text I am working from asks me to:
Show that if $f(x)$ and $g(x$) are odd functions, then $g(f(x))$ is odd.
Show that if $f(x)$ is an odd function ...
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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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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.
...
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What does $f(x)f(-x) \le 0$ mean when $f(x)$ is a cubic function?
Like the title, what does $\forall x\in \mathbb{R}, f(x)f(-x) \le 0$ say about $f(x)$, when $f(x)$ is a cubic function? The book says $f(x)$ has to be the odd function, but I can't figure out rigorous ...
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Proving that $y = a_{2n}x^{2n} + a_{2n−2}x^{2n−2} + . . . + a_2x^2 + a_0$ is an even function.
This question is from Larson's Precalculus book.
Prove that a function of the form
$$y = a_{2n}x^{2n} + a_{2n−2}x^{2n−2} + . . . + a_2x^2 + a_0$$
is an even function.
I don't understand how that ...
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The sine transform of symmetric functions
I am trying to think about the sine transform of symmetric functions, chiefly the unit rectangle and the unit triangle function. I am poor with LaTeX so I'll do this as best I can
Say I have the ...
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Finding Fourier series of $f(x)+c$ given that of $f(x)$
So, I have the function $f(x)$ over the interval $[-\pi,\pi]$ defined as under.
$$f(x)=\begin{cases}1+2x/\pi , -\pi\le x\le 0 \\
1-2x/\pi , 0< x\le \pi\end{cases}$$
The thing is computing the ...
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Determining if $f(x) = \frac{x}{x+1}$ is Even or Odd
This is likely a silly question, but I cannot seem to wrap my head around the following:
I am trying to determine whether the following function is even or odd: $f(x) = \frac{x}{x+1}$.
I pass $-x$ as ...
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Find $f(x)$ from two-variable equation
Let $f$ be a function which satisfies the equation
\begin{equation}
f(x-y+a)-f(x+y+a)=2f(x)f(y)
\end{equation}
where $a$ is a positive real number and for every pair of real numbers $x$ and $y$. ...
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Even, odd, or neither $y=\frac{1-\cos{x}}{1+\cos{x}}$
Determine algebraically whether the given function is even, odd, or neither.
$$y=\dfrac{1-\cos{x}}{1+\cos{x}}$$
I will substitute $−x$ into the function, and then simplify. $$y(-x)=\dfrac{1-\cos(-x)}{...
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Which even function $f(x)$ satisfies $\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)~dx=\frac{1}{n}$?
Question: Which even function $f(x)$ satisfies $$\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)~dx=\frac{1}{n}?
$$
Alternatively, which even function $f(x)$ satisfies $$\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\...
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Integrate $\int^\infty_0 \dfrac{\sin x}{x^3}\,dx$
By using this method we can evaluate $\displaystyle\int^\infty_0\dfrac{x-\sin x}{x^3}\,dz=\dfrac{\pi}{4}$ and I intended to solve $\displaystyle\int^\infty_0 \dfrac{\sin x}{x^3}\,dx=\displaystyle\int^\...
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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 ...
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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 ...
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Odd function with no critical points must be linear.
Let $f: \mathbb{R}\to \mathbb{R}$ be a differentiable odd function, i.e., $f(x) + f(-x) = 0$ for all $x\in \mathbb{R}$ and $f^\prime(x)$ exists for all real $x$. Further assume the followings:
$
\...
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How to transition from approximation of odd function using $\sin$ and $\cos$ to only using $\sin$?
I'm failing to understand the following transition:
Given that $f$ is an odd function defined on $[-\pi,\pi]$, we know that for all $\epsilon>0$ there exist $a_n, b_n \in \mathbb{R}$ such that:
$$||...
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Obtain a function by it's even or odd part
Since every function can be divided in a even and a odd part:
\begin{equation}
f_e(x)=\frac{f(x)+f(-x)}{2}
\end{equation}
\begin{equation}
f_o(x)=\frac{f(x)-f(-x)}{2}
\end{equation}
\begin{equation}
f(...
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Derivation of function$~g(x)=f(x)+f(-x)~$is an even function where$~f(x)~$is an arbitrary continuous function.
Let$~f~$be any function which is defined for all numbers. Show that$~g(x)=f(x)+f(-x)~$is even.
$$\operatorname{e.g.}~f(x)=x^2\implies g(x)=x^2+(-x)^2=2x^2\leftarrow~~\text{even}\tag{1}$$
$$\...
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