# Tagged Questions

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### $\int_{-1}^{1} x^{k+i} P_n(x)dx$, $P_n$ Legendre polynomial.

I was wondering whether there is a way to say what $$\int_{-1}^{1} x^{k} P_n(x)dx$$ is, where $k,n$ are positive integers or zero and $P_n$ is the n-th Legendre polynomial? I am looking for an ...
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### Riemann Integrating a Step Function

So I've been trying to prove a step function with countably infinite discontinuities is Riemann integrable using only properties of Riemann integration, no Lebesgue or gauge integration for example. ...
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### I need to show that this sequence is increasing and I'm almost there but I need help on last step.

Let $(1+\frac{1}{n})^n$ be a sequence and $f(x)=(1+\frac{1}{x})^x$ on $[1,inf)$. I need to show that f is non-decreasing by showing that $f'(x)\ge0$. So far I have: Let $g(x)=ln(f(x))$, where $ln$ ...
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### Computing $\int_{\gamma} {dz \over (z-3)(z)}$

Compute, using the Cauchy Integral Formula, $$\int_{\gamma} {dz \over (z-3)(z)}$$ where $\gamma$ is the circle of radius $2$ centered at the origin, oriented counterclockwise. ...
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### Finding all continuous solutions to an integral

I need help with both parts of this problem. Part (i) seems obvious, because the integrand $f(t)$ would become $F(t)$, which is obviously differentiable because its derivative is $f(t)$ by ...
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### Prove by using step functions: $\int_{-b}^{b}\sin(x)\ dx = 0$

The Assignment: Let $b > 0$. Prove by using step functions: $$\int_{-b}^{b}\sin(x)\ dx = 0$$ The claim itself is obvious, but I have no idea how to prove it with step functions. My idea was ...
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### Higher-dimension integrability (over rectangles) well-defined

Here is the problem and my work toward a proof: Question: Prove that in the following definition, the value of $\int_E f dx$ is independent of the choice of rectangle $J$: Definition: ...
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### Variation on Fubini's Theorem

My attempt: Let $P_1$ be a regular partition of $R_1$ and $P_2$ a regular partition of $R_2$. Denote by $P$ the corresponding regular partition of $R_1\times R_2$. Given a generalized rectangle ...
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### Definition of a Regular Partition of a Closed Generalized Rectangle in $\mathbb R^n$

What the heck does this definition of a regular partition $P$ of $R$ mean? I follow what it is saying until we get to the last part, "the $k_1\cdot k_2\cdot \cdots \cdot k_n$ subrectangles of the ...
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### If $f \in C^\infty$, and $f$ is nonnegative and integrable in $\mathbb{R}$, can I say that $f^\prime$ is integrable?

I'm not sure how to describe the question any further in the title than it is, but I will try to explain what I have done. If $f$ is a Schwartz function, I believe that $f^\prime$ will always be ...
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### proof of an relation

Let $\Omega = \mathbb{R}^2_+=\{(x,y)\in \mathbb{R}^2; y>0\}$ et soit $v \in H^1_0(\Omega)$ and let $h \neq 0$. Let $$D_h v = \dfrac{v(x+h,y) - v(x,y)}{h}$$ The questions are: 1- Prouve that ...
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### Riemann-integrable functions and pointwise convergence

Hello, I was hoping for some advice on finding a function which will satisfy this. I think I am okay with the actual execution of the answer, but I don't know how I'm supposed to find a suitable ...
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