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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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
$$
\begin{aligned}
Ci(x) &:= - \int_x^\...
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votes
0
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views
Reference Request: Definition on parity of scalar-valued multivariate functions (even or odd)
I am studying the densities of multivariate Gaussian distributions and I am curious about the parity of those scalar-valued multivariate functions.
For a function $f(\mathbf{x}): \mathbb{R}^n \...
- 25
2
votes
1
answer
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Translation of odd and even functions
Let $\varphi: \mathbb{R} \to \mathbb{R}$ be a periodic function of period $L>0$, that is,
\begin{equation}\label{periodicitycondition}
\varphi(x+L)=\varphi(x),\; \forall\; x \in \mathbb{R}. \tag{1}
...
- 1,657
0
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1
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Even/Odd with reference to the interval (domain)
As the definition goes, a function $f(x)$ is even if $f(-x)=f(x)$ and it is odd if $f(-x)=-f(x)$, in which the domain is not paid enough attention to.
For example, $f(x)=x^2$ is even for any symmetric ...
- 379
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Anything interesting known about this generalization of even and odd functions?
Let $n \in \mathbb N$. Let's say a complex function $f: U \rightarrow \mathbb C$ is "of type $k \pmod n$" if for one (and hence every) primitive $n$-th root of unity $\omega$,
$$f(\omega z) =...
- 27.1k
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1
answer
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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 ...
- 103
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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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0
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25
views
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 ...
- 356
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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)...
- 103
2
votes
1
answer
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views
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 ...
- 195
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1
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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 ...
- 43
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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 ...
- 1,685
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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 ...
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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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