# Questions tagged [even-and-odd-functions]

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 $f$...
### 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 ...