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Questions tagged [even-and-odd-functions]

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0
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1answer
39 views

f(x) = g(x) + i h(x) if g(x) is real even function, h(x) is real odd function how about the root of f(x)=0

I want to know whether f(x)=0 always has complex root if real part of f(x) is a real even function and imaginary part of f(x) is a odd function. Thank you very much!
0
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0answers
31 views

Is there some way to prove the imaginary part of any complex function is even or odd function?

I have a complex function as f(x) = g(x) + I h(x). Because f(x) is very complicated function and depend on some parameter. I want to know whether there is a way to prove that g(x) or h(x) is even ...
10
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1answer
128 views

Generalization of even / odd functions

The following four examples all have a similar structure: Every function $f:\Bbb R \to \Bbb R$ has a unique decomposition $f = f_e + f_o$ where $f_e$ is an even function ($f_e(-x) = f_e(x)$) and $...
2
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2answers
32 views

Partial fraction decomposition of even function

I need to do partial fraction decomposition of this function (to solve its integral): $\frac{t^2}{t^4+4}$ Since $t^4+4=(t^2+2t+2)(t^2-2t+2)$ I would do: $\frac{t^2}{t^4+4}=A\frac{2t+2}{t^2+2t+2}+B\...
1
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3answers
62 views

Is the function $z=x^{3}y$ even or odd?

I just wanted to know because I am trying to calculate $\int_{-1}^{1}\int_{-1}^{1}x^{3}ydxdy$.
1
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3answers
69 views

An antiderivative of an odd function is even. Proof in general.

We have: $\int f(x)dx=F(x)+C$, $f(x) = -f(-x)$ - odd function. Proof that $F(x)$ - even function. My suggetion (please, check it): Since $\int f(x)dx = F(x) + C$, $\int f(-x)dx = -F(-x) + C$. ...
2
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0answers
83 views

Can a function $R^2 \rightarrow R^2$ be both even and odd for each of its variables?

It is clear that a function $R \rightarrow R$ that is both even and odd must be identically zero. I'm interested in how this could or couldn't be generalized to $R^2 \rightarrow R^2$ functions. I ...
1
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1answer
20 views

Determining if a function is even or odd using a system of equations and solving for unknown constants

I'm following a solution to a problem but I wanted to ask about a particular step. I have the following equation $$ g(y)f'(x)=(C_1\cos(\sqrt{G}ky)+C_2\sin(\sqrt{G}ky))(C_3\sinh(kx)+C_4\cosh(kx)) $$ ...
1
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0answers
30 views

Generalization of even-odd parity (symmetry) decomposition of 2-dim functions

For any "well-behaving" real-valued (1-dim) function $f(x)$, the following decomposition that can always be done is basic and at the same time very useful. $$f(x) = f_{ev}(x) + f_{od}(x) \qquad \begin{...
4
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2answers
52 views

Regarding whether two functions are even or odd

I am solving the integral $$\int_\frac{-π}{2}^\frac{π}{2} x \cos x+ \tan x^5 \,dx$$ In the solution, both $x \cos x$ and $\tan x^5$ are said to be odd functions. Are these functions even or odd? ...
0
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1answer
22 views

Integral of an even function using symmetry

$$\int_{-99}^{99} (ax^3+bx^2 + cx) \,dx= 2\int_{0}^{99} bx^2 \,dx$$ Is this assertion true or false? I would say this is only true if $a$ and $c$ are zero, so that $bx^2$ is even. But this is false ...
0
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1answer
25 views

How do i show that this function :$f(x)=\alpha x +\beta+ \frac{\gamma}{\alpha'+\beta' x}$ always have symetric point?

let $f$ be a real valued function defined as $f(x)=\alpha x +\beta+ \frac{\gamma}{\alpha'+\beta' x}$ , I want a simple method to show that every function of the precedent form always has a symetric ...
2
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3answers
204 views

Determining properties of a polynomial $f$ satisfying $f(x^2)-xf(x) = x^4(x^2-1)$ for $x \in\Bbb R^+$

Let $f$ be a polynomial satisfying $f(x^2)-xf(x) = x^4(x^2-1), x \in\Bbb R^+$. Then which of the following is correct? A) $f$ is an even function B) $f$ is an odd function C) $\...
1
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1answer
32 views

An even function, $f(z)$, analytic near 0 can be written as another analytic function, $h(z^2)$

If f is an even function, $f(z) = f(-z)$, and is analytic near 0, then there exists a function h, also analytic near 0, such that $f(z) = h(z^2)$ I suspect this statement is true because the ...
0
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0answers
25 views

How to determine whether a function is even or odd in case the function has discontinuity at the origin?

$ \sqrt {(1+a^2/x^2)} =>\frac1 x\sqrt{(x^2+a^2)}$ The first expression is even (i.e remains same when we put -x in place of x), while the second one is odd. What am I doing wrong while going from ...
1
vote
1answer
22 views

Solution for $u_{t} = \alpha^{2}u_{xx}$ (problem with Fourier Series of inicial condition)

Solve the partial differential equation $u_{t} = \alpha^{2}u_{xx}$ with conditions: \begin{cases} u(x,0) = f(x)\\ u(0,t) = u_{x}(L,t) = 0 \end{cases} where $$f(x) = \begin{cases} \frac{2}{L}x,&...
3
votes
1answer
46 views

$F(x)=\int_{-2}^{2} dy f(x,y)$ is an even function, is $G(x)=\int_{-2}^{2} dy [f(x,y)]^2$ even?

I have a real valued function in two real variables $f(x,y)$ which is essentially a black box. The only thing I really know is that $$ F(x)=\int_{-2}^{2} f(x,y) dy $$ is even and that $$ \int_{-\infty}...
1
vote
1answer
51 views

Is $\ln\left( \frac{1+2x}{1-2x}\right)$ an odd or even function?

Is $$f:\left(-\frac{1}{2},\frac{1}{2}\right) \to \mathbb{R}, \ x \mapsto \ln\left( \frac{1+2x}{1-2x}\right)$$ an odd or even function? The function can be decomposed this way: $$f(x)=\ln(1+2x)-\ln(1-...
0
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3answers
29 views

Considering that $a$ and $b$ are odd functions, prove that the solution of the differential equation $y' + ay = b$ is even

$a$ and $b$ are two odd functions, continuous on R. How can I prove that the solution of the differential equation $y' + ay = b$ is even. I tried to find the solutions of the differential equation ...
0
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0answers
44 views

Odd and Even Functions for equations I don't know the exponents of

I'm solving a problem to which the final obstacle (I think) is knowing if this function is even or odd: Given $$xQ(x+2018)=(x-2018)Q(x)$$ and that $Q(1)=1$, what is $Q(2017)$? My best attempt so far ...
-4
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1answer
265 views

If the function is neither even nor odd ,is its fourier series coefficient a complex? or is this wrong? [closed]

i thought if the fourier series coefficient of $f$ is neither even nor odd,$f$ must be the complex. we can know the function,$f$, is real or imaginary or complex from its FS coefficients. Now i don'...
0
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3answers
49 views

Prove $(\sin x)^{1/3}$ is an odd function

A question asks to prove that $f(x)=(\sin x)^{1/3}$ is an odd function. I have begun by using the idea that for a function to be odd, $f(x)=-f(-x)$, such that... $$-f(-x)=-[\sin(-x)]^{1/3}$$ $$=-[-\...
14
votes
1answer
209 views

If $f \circ f$ is odd, then is so $f$?

It is straightforward to see that $f \circ f$ is odd whenever $f$ is odd. Indeed, assuming $f(-x) = -f(x)$ for all $x$, we get $$ f(f(-x)) = f(-f(x)) = -f(f(x)). $$ Hence, $f \circ f$ is an odd ...
1
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1answer
57 views

The domain of $f(x) = a^x + a^{-x}, a>0$.

The domain of $f(x) = a^x + a^{-x}, a>0$. My attempt : Since the domain of the exponential function is $\mathbb{R}$, and since our function is the sum of two exponential functions and since the ...
0
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1answer
25 views

How to show a continuous random variable with even pdf also has a even moment generating function?

The moment generating function is $M_X(t)=$$\int_{-\infty}^{\infty} e^{tx}f(x) dx$ if X is continuous.
0
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0answers
34 views

Characterizing two even polynomials generating the unit ideal

Let $f=f(t),g=g(t) \in k[t]$ be two even polynomials, namely, each is a sum of even degrees monomials (including degree zero). $k$ is a field of characteristic zero. Assume that we know that there ...
0
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0answers
79 views

Antithetic sampling and Monte Carlo simulation

Consider: \begin{align} f(x) = \left\{ \begin{array}{ll} 0, & 0 < x < 0.9 \\ 100, & 0.9 < x < 0.91 \\ 0, & \textrm{otherwise} \\ \end{array}\right. \end{align} Determine ...
0
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1answer
84 views

Fourier transform of $\sin(t)$ for range $0$ to $\pi$

I have find out the Fourier transform of the $\sin(t)$ for range $0$ to $\pi$. Graph is added in the link. https://www.mathcha.io/editor/p2g4u6yI3nHX6hWk As $\sin(t)$ is a real and odd function, can ...
0
votes
1answer
70 views

What's the connection between even/odd harmonics and even/odd (transfer) functions?

What's the connection between even/odd harmonics and even/odd (transfer) functions? Why do they correspond to each other? Particularly, if the transfer function is/contains even or odd functions, ...
3
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1answer
79 views

Limits of ${\int\limits_{-\infty}^\infty f(x)e^x\mathrm{d}x}$ changed to $(0,\infty)$ if $f(x)$ is even

In a medical imaging paper the following integral is changed from: $$\int_{-\infty}^\infty f(x)e^x\mathrm{d}x$$ To: $$\int_0^\infty2f(x)\cosh(x)\,\mathrm{d}x$$ I understand how you could get $2f(x)$ ...
0
votes
3answers
63 views

If $f(x)$ is odd, then prove that $f'(0)=0$

If $f(x)$ is odd, then prove that $f'(0)=0$, assuming that $f'(0)$ exists. We have that $f(x)$ is odd. Therefore, $f(-x)=-f(x)\implies -f'(-x)=-f'(x)$. What I do after this?
5
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0answers
73 views

N-dependent (even) function integral

We want to compute, for any $n \in \mathbb{N}$ the following integral: $$\int_{-1}^{1} \frac{x^n}{\sqrt[n]{1+x}+\sqrt[n]{1-x}}dx$$ My attempt: if $n$ is odd, the integral is trivially equal to $0$ ...
5
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2answers
570 views

Area under a curve of an odd function from negative infinity to positive infinity

In integration, there is a property that says: If you're integrating from -a to a some odd function f(x), then the area under the curve between -a and a is zero. I was listening to this in class , ...
0
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1answer
15 views

What does it mean for an odd function to be odd about an end of an interval (at $x=L$)?

What does it mean for an odd function to be odd about an end of an interval (at $x=L$, when the interval is $[0,L]$ or $[-L,L]$)? E.g. the sine function is odd under reflection about $0$, but also ...
29
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4answers
2k views

Can we use symmetry rules in improper integrals?

I wish to evaluate the integral $$I=\int^{\infty}_{-\infty}xe^{-x^2}dx$$ Can I simply note that that $f(x)=xe^{-x^2}$ is an odd function and say $I=0$? The only reason I have doubts is because of ...
1
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3answers
116 views

Is it possible for the sum of even and neither odd nor even function to be even or odd?

Consider the functions $f(x)$, $g(x)$, $h(x)$, where $f(x)$ is neither odd nor even, $g(x)$ is even and $h(x)$ is odd. Is it possible for $f(x) + g(x)$ to be even; odd? For the second case I can ...
5
votes
1answer
62 views

is there any maping from $S^1$ to $S^1$ of odd degree, which is not an odd function?

I want to prove for every continous function from $s^n$ to $s^n$ of odd degree there exists $x$ such that $f(-x)=-f(x)$ so I used this "that the sum of two functions of odd degree must be odd" but I ...
1
vote
1answer
30 views

Probability - can't understand odd function (random variable prob.)

PROBLEM PICTURE Hello, I do not understand above pictures's "ODD FUNCTION => 0" part. I need more detailed explanation on how it becomes to be 0. Thank you very much.
0
votes
1answer
45 views

Number of odd function over symmetric interval.

Let $ A = \left \{ -2,-1,0,1,2 \right \} $ How many $ f : A \rightarrow A $ odd functions exist? How am I supposed to approach these type of problems? The graphic representation doesn't seem to ...
0
votes
2answers
61 views

Result of 1+1 in odd digits number system [closed]

Consider number system which uses only digits ${0,1,3,5,7,9}$ (odd digits with the except of 0). What will be the result in such system of summation 1+1 ? As I ...
1
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2answers
3k views

Prove that any function can be written as the sum of an even function and an odd function. [duplicate]

I understand some of the basic concepts that surrounds even and odd functions but this question just stumped me and I'm not sure on how to tackle it. Any Starting points/methods would be helpful ...
0
votes
1answer
37 views

Counterexample to: If odd Taylor coefficients for g in a=0 equals 0, then g is an even function. [closed]

Let g be an infinitely differentiable function. Find counterexample to: If the odd Taylor coefficients for g in a=0 equals 0, then g is an even function.
1
vote
1answer
43 views

Example on function g that is not even and where the derivative fulfill g^(2n-1)(0)=0

I am stuck in finding a function $g(x)$ that is not even, so $g(x)\neq g(-x)$, can be differentiated infinity many times and has $$\frac{g^{(2n-1)}(0)}{(2n-1)!}=0.$$ I have tried with at lot of ...
0
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0answers
39 views

What does it mean for an initial boundary condition to be even or odd?

So the boundary conditions for the solar theory question I am looking at are: $$B_z (x,0) = \begin{cases} B_0 , & -a \leq x \leq a \\ 0, & -L \leq x < -a, \ a<x \leq L \end{cases} $$ I ...
3
votes
1answer
164 views

Is there such a thing as an Even Matrix?

An even function is one in which $f(x)=f(-x)$. For two variables I believe this is $f(x,y)=f(-x,-y)$ If I wish to make a 2D even matrix how would I do this? $$ \begin{matrix} (0,0) & (0,1) \\ ...
-4
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1answer
40 views

This question blow up my mind pls someone explain this [closed]

!This concept is taken from mathematical circle Pls try to explain as easy you can
-1
votes
1answer
22 views

Can we draw closed path made up of 9 line segments each of which intersects exactly one of the other line segmentsn

This problem is taken from MATHEMATICS CIRCLE book Can anyone tell me how the figure of this problem look like Pls elaborate with simplest explanation
3
votes
1answer
40 views

Find the range of $a$

If the function $$f(x)=[4.8+a\sin x]$$ where $[.]$ denotes the greatest integer function, is an even function then find the range of $a$. So, if the function of even then $f(-x)= f(x)$. Therefore,$$[...
3
votes
0answers
66 views

For Odd continuous functions from s2 to R there exist x s.t. f(x)=g(x)=0

For two odd continuous functions $f$ and $g$ both from $S^2$ to $\Bbb R$ , there is a $x$ such that $f(x)=g(x)=0$. Please someone give me a hit about this...
-3
votes
2answers
63 views

If $\int_{-4}^4 f(x)(\sin x +1)\, dx = 8, \; \int_{-2}^4 f(x)\, dx = 4$ where $f(x)$ is an even function, what is $\int_{-2}^0 f(x)\mathrm dx\ ?$ [closed]

If $$\int_{-4}^4 f(x)(\sin x +1)\mathrm dx = 8, \quad \int_{-2}^4 f(x)\mathrm d x = 4$$ where $f(x)$ is an even function, then what is the value of $$\int_{-2}^0 f(x)\mathrm dx\ ?$$ The answer is $0$....