Questions tagged [even-and-odd-functions]

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39 views

Evaluate the Improper Integral(help) [closed]

I encountered the following integral while solving a log-normal distribution question. Initially, I thought since its a odd function, it evaluates to zero. But I think, since its a improper integral, ...
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2answers
80 views

Evaluating $\int_{-a}^a x^{2n+1}\mathrm{d}x$ for all non-negative integers $n$ simultaneously

My assumption would be $$\int_{-a}^a x\ dx=0$$ Am I on the right track here? Also, for indefinite integrals $$\int (f)x\ dx$$ would this be correct as well? Background My professor raised this ...
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2answers
45 views

If $\int_{-1}^1 fg = 0$ for all even functions $f$, is $g$ necessarily odd?

Suppose for a fixed continuous function $g$, all even continuous real-valued functions $f$ satisfy $\int_{-1}^1 fg = 0$, is it true that $g$ is odd on $[-1,1]$? My intuition is telling me that this is ...
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3answers
61 views

If $f$ is odd and periodic then a translation of $f$ is even?

Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a odd and periodic function, with period $L>0$. If we define $$g(x):=f\left(x-\frac{L}{2}\right), \; \forall \; x \in \mathbb{R},$$ then $g$ is ...
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1answer
23 views

Does there exist a continuous and differentiable EVEN function whose slope at zero isn't zero?

as I understand, there should not be a case, where the slope at x=0 is nonzero, if the function is even and continuously differentiable at all points including x=0.
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5answers
68 views

Determining when a function is neither even nor odd

Suppose that $f(x) = \dfrac{x}{x+1}$. I need to determine if it is even, odd or neither. Now $f(-x) = \dfrac{-x}{-x+1} = \dfrac{x}{1-x} \text{ and } -f(x) = \dfrac{-x}{x+1}$. I can "see" ...
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3answers
56 views

Even and odd functions for Taylor serie

I have asked this question before, sorry, but I'm still confused about how I can show it. Hope anybody can help me? We let $f:\mathbb{R}\to\mathbb{R}$ be infinitely often differentiable function and ...
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2answers
48 views

If even function then … [duplicate]

We let {$a_n$}$_{n\in N}$ be $a_n$=$\frac{f^{n}(0)}{n!}$. I have to show that if $f$ is an even function so is $a_{2n-1}$$=0$ for all n$\in$N. How can I show it? By induction maybe? Can anyone give a ...
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3answers
38 views

Cosine is an even function, then why does it have a negative value in 2nd and 3rd quadrants?

Cosine is an even function, then why does it have a negative value in 2nd and 3rd quadrants respectively?
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1answer
10 views

Fourier Series Using the Odd and Even extensions

For the function $f(x)=x$ where x belongs to [0,$\pi$], I have found the Fourier series for $f_e(x)$ and $f_o(x)$. If I wanted to find the Fourier Series of the function, can I add the series ...
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1answer
41 views

commenting on whether $f'(x)$ is even or odd.

The question is as follows: Let $f(x)$ be a differentiable function $\forall x,y \in \Bbb R$ and $$f(x-y),f(x),f(y),f(x+y)$$ are in AP then comment whether $f'(x)$ is even or odd (given $f(0) \neq 0)...
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2answers
245 views

If $g(x)$ is an even function with $f(0)=0$, then how can we deduce that $f(x)$ is an odd function?

Let $f(x)=\int_0^x g(t)\,\mathrm dt$, where g is a non-zero even function. If $f(x + 5) = g(x)$, then $\int_0^xf(t)\,\mathrm dt$, equals:$$\int_{x+5}^5g(t)\,\mathrm dt,\quad2\int_5^{x+5}g(t)\,\mathrm ...
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2answers
55 views

If $a(x)a(-x)=b(x)$, who is $a(x)$?

If we have $a(x)a(-x)=b(x)$, how can we find $a(x)$? Or, if we have $c(x)+c(-x)=d(x)$, how can we find $c(x)$? Please note that these functions should be continuous ones. For example, if $a(x)a(-x)...
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2answers
150 views

How to find distinct n numbers with following conditions? [closed]

Let $N$ be an even nonnegative integer. How to find $N$ distinct integers with following conditions? The first $N/2$ integers are even. The remaining $N/2$ integers are odd. The sum of the ...
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0answers
10 views

Wave equation: function 2L-periodic.

I have to prove that any function that is even with respect to x=0 and x=L is necessarily 2L-periodic. We are studying wave equations but I don´t know how to prove it. Can you give me a clue?
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0answers
8 views

Are all odd harmonic functions are half-wave symmteric?

I know that the Fourier series expansion of a half-wave symmetric function contains only odd harmonics. Is the reverse true?
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1answer
23 views

Why does extending a function to an even/odd function work when finding a Fourier series?

A function $f(x)$ over an interval $[0,l)$ can be extended to an odd/even function. E.g. the function $f(x) = x$ can be extended to an even function $ g(x) = \begin{cases} x & 0 < x \\ -x &...
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1answer
13 views

Proving the composition of an even function and an even function is even using the given definitions

There are multiple answers to this question, but they use a slightly different version of what it means to be an odd function. I wanted to know what I did wrong in my process of trying to arrive at a ...
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4answers
46 views

Why does the lower limit of Integration change to a zero and where does the 8 come from?

I'm working through an example from Stewart's book, and I can not work out why the lower bound of integration changes to a $0$. Also I am confused as to where the $8$ comes from? I am assuming there ...
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1answer
19 views

polynomial even function $f(-x)f(x)$ [closed]

How to prove: $f(-x)f(x)$ is an even function.
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1answer
28 views

Even Odd Functions

Let $e(x)$ be an even function and let $o(x)$ be an odd function, such that $e(x) + x^2 = o(x)$ for all $x.$ Let $f(x) = e(x) + o(x).$ Find $f(2).$ We could write $f(x)=e(x)+e(x)+x^2.$ We want to ...
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0answers
24 views

Infinite Grid problem

An infinite square grid(made of square blocks)  contains integer coordinates(points with integer coordinates)  having edge length equal to a. Your goal is to find out what number of moves is not ...
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0answers
23 views

The inverse of Schwarz-Christoffel mapping of the upper half-plane is odd

I am reading Ahlfors' complex analysis. He gave this function $F(z)=\int_0^z \frac1{\sqrt{1-t^2} \sqrt{1-k^2t^2}}dz$ which conformally maps from the upper half plane to a rectangle $R_0$, here $0<...
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0answers
22 views

“Composition of an even and an odd function is even” means both $f(g(x))$ and "$g(f(x))$ are even?

Wiki composition of an even function and an odd function is even Is "composition order" significant here? Let $f(x)$ - odd, $g(x)$ - even It means only $f(g(x))$ is sure to be even? Or both "$...
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0answers
22 views

Integrable even function that is not necessarily continuous without using partitions.

fellow mathematicians. I was given the following exercise: Let $f:[-a,a]\to\mathbb{R}$ an integrable function s.t $a>0$. Prove the following: 1.- If $f(x)$ is an even function: $$ \int_{-a}^{a}{f(...
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1answer
86 views

Integrating odd Legendre polynomials using generating function

I must show using generating function of Legendre polynomials, that \begin{align} \int_0^1 P_{2n+1}(x)\phantom{1}dx = (-1)^n\frac{(2n)!}{2^{2n+1}n!(n+1)!} \end{align} My attempt is to change the ...
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1answer
31 views

Fourier series representation of a square wave inverter waveform [closed]

I am trying to find the Fourier series representation of a inverter output square wave waveform as shown in attached figure. Finally I get certain components being cancelled out and the only ...
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0answers
42 views

Does $\int_{-1}^1\frac{1}{x}dx$ converge or diverge? [duplicate]

Does $\int_{-1}^1\frac{1}{x}dx$ Converge or diverge? Is the following a valid proof of convergence?: $$= \lim_{a\to0+}(\int_{-1}^{-a}\frac{1}{x}dx + \int_{a}^1\frac{1}{x}dx)$$ $$= \lim_{a\to0+}([\ln|...
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1answer
57 views

Solutions to $f''(x)=-f(x)^2$

Suppose that $f''(x)=-f(x)^2$ and $f(0) = f(1) = 0$. How does one show that $f(1-x)=f(x)$? Is it sufficient to show that $f'(0.5)=0$? I'm also curious about the number of solutions to this ...
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2answers
57 views

Show that if f(x) is an even function, then g(f(x)) is an even function.

The question does not say whether g(x) is odd, even or neither. Proving that g(f(x)) = g(-f(x)) (Proof that g(f(x)) is even) I swapped f(x) with f(-x) So that g(f(-x)) = g(-f(x)). But from there, ...
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1answer
55 views

Improper integral of an odd function

would $\int\limits_{-\infty}^\infty f(x)\ dx=0$ always be true if $f$ is an odd function?
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1answer
51 views

Both even function and odd function [duplicate]

I know that 0 is an even function and an odd function. How can I prove f is both even and odd if and only if it is the constant 0 function
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3answers
45 views

Confusion about the definition of an even function for $f(x) = x^{4\over5}$.

I am confused if the function $f(x) = x^{4\over5}$ is an even function. If we only consider the real roots, then the function is even in the sense that $f(-x) = f(x)$. However, since $x^{1\over 5}$ ...
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2answers
49 views

Is there a continuous odd function from $S^3$ to $S^1$?

Let $S^3$ and $S^1$ be the 3-sphere in $\mathbb{C}^2$ and the circle in $\mathbb{C}$, respectively. Is there a continuous map $f: S^3 \to S^1$ where $-f(z_1,z_2) = f(-z_1, -z_2)$ for all $(z_1,z_2) \...
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1answer
41 views

Is a function $f(x)=\ln({x^2-1})$ even and symmetric

We have a function: $$ f(x)=\ln(x^2-1) $$ The function is symmetric because: $D_f=(-\infty,-1) \ \cup\ (1,\infty)$ I understand this as if we would multipy this by $-1$ we would get the same $D_f$ ...
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2answers
51 views

Determine if function $f(x)=\ln \left ( \frac{x+1}{x-1} \right )$ is odd or even.

Determine if function $f(x)=\ln \left ( \frac{x+1}{x-1} \right )$ is odd or even. My solution: $$ \begin{align} f(-x)&=\ln\left ( \frac{-x+1}{-x-1} \right )\\ &=\ln \left ( \frac{-(x-1)}{-(x+...
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2answers
31 views

Continuous, strictly monotonic and odd functions [closed]

Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous, strictly monotonic and odd. Is $f$ necessarily a polynomial function?
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2answers
44 views

Is $\frac{(|x| - x^2)}{\sin x}$ an odd function, if it is odd, how do I prove that?

This is the function: $$f(x)=\frac{|x|-x^{2}}{\sin x}$$ I am not understanding where to go from the second line: $$\begin{array}{l} f(-x) & =\displaystyle\frac{|-x|-(-x)^{2}}{\sin(-x)}\\ \...
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2answers
31 views

How do you see this function is odd?

I thought I am supposed to do check $f(t)=-1$ and compare with $f(-t)$ $f(-t)=-1$ If $f(t)=f(-t)$ the function is even. But this function is odd.
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1answer
21 views

Question about the solution to a Fourier-cosine function

I have this function $$ f(t) = \begin{cases} -t, & \text{0 ≤ t ≤ π} \\[2ex] \end{cases}$$ I'm asked to show solve the Fourier-cosine function After setting up and solving the integrals I get ...
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1answer
66 views

What is the reason for the shape of these Fourier series graphs?

I've been working on understanding Fourier analysis both with pencil and paper as well as on how to write an algorithm that computes fourier coefficients (yes, I know there are already libraries for ...
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1answer
100 views

Are bump functions even within their compact support?

Assume that $V$ is a real vector space of smooth non-analytic functions with a compact support $[a, b]$ (i.e vector space of bump functions) and $f \in V$. I'm trying to show that: $$I=\int_{b}^{a}{f(...
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2answers
38 views

Please help me understand the solution to this graph

I have a function $$f(t) = \begin{cases} 0, & \text{0≤t<1} \\[2ex] 2, & \text{1≤t<2} \end{cases}$$ According to the solutions manual, the period is 2 this is an odd function so I add ...
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3answers
58 views

Does real characteristic function imply even density

Does real characteristic function imply even density? So, I was wondering over this. I know that when a characteristic function is real, it is even. Also, I know this result when the characteristic ...
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3answers
220 views

Integral using even function

$$\int_{-{2 π}/3}^{{2 π/3}}-\frac {4\cos(x)}{1 + e^{-4x}} \,\mathrm{d}x $$ I have tried writing this as an even function so I can evaluate: $$\int_0^{{2π}/3} f_\text{even}(x)\mathrm{d}x$$ But I'm ...
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1answer
28 views

Is an odd/even function multiplied by $i$ still an odd/ even function?

Let's say I have a function that I want to integrate an even function multiplied by $i\sin(x)$ between $-1$ and $1$ and the function is $1-|x|$ then does this integral become zero because integrating ...
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1answer
51 views

Creating an odd function $f(x)$ such that $g(x) = f(x - 18)$ is an even function

I am tasked with finding an example of a function $f(x)$ such that the function $g(x) = f(x-18)$ is an even function. I understand even and odd functions. However, I am unsure how to create an odd ...
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1answer
52 views

How can a function defined on symmetrically placed interval be written as sum of an even and odd function?

I know how to find Fourier series of a function,but i found following question and I stuck. "Show that any function $f(x)$ defined on symmetrically placed intervals can be written as sum of an ...
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0answers
58 views

Integral is zero; is the integrand necessarily odd?

As we know, the integral of an odd (integrable) function over a symmetric interval disappears. Now, I am having difficulties about proving a somewhat converse statement of this, specifically: Let $f$...
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5answers
163 views

Is $\sqrt{x^2}$ even or odd?

Is the function $x\mapsto \sqrt{x^2}$ even or odd? Mentioning the square root does not have negative sign, $\sqrt{x^2} = \pm x$ As it is clear LHS is even and RHS is odd for both sign, which one is ...