Questions tagged [calculus]
For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.
138,147 questions
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Changing the order of integration in an iterated integral with a single varible function
I am trying to consider a double integral:
$$
\int_t^\infty \int_s^\infty f(r) dr ds <+\infty
$$
where $f:\mathbb{R} \to \mathbb{R}$ is a smooth function, but NOT a non-negaitive function. And the ...
- 37
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0
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74
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Proof of Euler identity using ODE
I am writing an article to prove Euler identity :$e^{i\pi}+1=0$
Here the main part:
Consider the function :$ \mathbb{R} \rightarrow \mathbb{C}, f(x)=e^{ix}$
Differentiating twice,we get : $f''(x)=-f(x)...
- 58
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0
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30
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Is my Euclidean-style proof valid? It is for a summation of infinitely many line segments equaling to a finite length without calculus, or limits. [closed]
I have attempted to prove that the sum of infinitely many quanities can still equal a finite quantity without using calculus, measure theory, or any other modern mathematical tool such as set theory.
...
0
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1
answer
32
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Base of the line integration [closed]
Does line integral integrate over the projection of a 3d curve onto the x-y plane or over the 3d curve itself as the base of the integration?
Thanks
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96
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Getting different answers for integration problem: $\int_0^2 x d( \{x\} )$
My teacher used integration by parts to solve the problem like so:
$$\int_0^2 xd(\{x\})
=[x\{x\}]_0^2-\int_0^2 \{x\}dx\\
=0-\int_0^1 xdx-\int_1^2 (x-1)dx$$
which comes out to -1. But when I was ...
1
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0
answers
108
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How to integrate $\int \frac{4x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{d}x$ analytically?
I'm trying to solve the integral
$$\int \frac{4x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{d}x$$
I do know that a similar integral
$$\int \frac{12x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{...
- 13
1
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1
answer
89
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True or false? If $\lim n|x_n-x_{n+1}|=0$ then $\{x_n\}$ converges. [duplicate]
I want to know whether it is true that if a real sequence $\{x_n\}_{n=1}^\infty$ satisfies $\lim\limits_{n\to\infty} n|x_n-x_{n+1}|=0$ then it converges.
I guess it is false but I can't find a ...
- 157
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2
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124
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Can I treat the limit as a constant? [closed]
Imagine that the limit as $h$ approaches infinity of $f(1 + hx)$ is $g(x)$.
$$\lim_{h\to \infty} f(1 + hx) = g(x)$$
Can I then say that $f(x)$ is equal to the limit as $h$ approaches infinity of $g\...
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1
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Is my understanding correct about the inverse relationship between derivatives and integrals?
I've learned that derivatives and integrals are inverse operators, but am not completely sure why. I've looked at many resources to understand why, and here goes.
The integral gives a sum of the total ...
- 47
2
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1
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56
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Find out the distance between centers of two intersecting semi-ellipses $x^2/a^2+y^2/b^2=1$.
There are two identical semi-ellipses, one with center at the origin $O$, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, and the other at $R$, $\frac{(x-d)^2}{a^2}+\frac{y^2}{b^2}=1$.
Find out the distance $d$ ...
- 1,290
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24
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How to setup bounds for triple integrals [closed]
Set up (do not evaluate) triple integrals in spherical coordinates in the orders dρdϕdθ and
dϕdρdθ to find the volume of the cube cut from the first octant by the planes x = 1, y = 1
and z = 1.
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4
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Dummy variable rule for indefinite integrals?
My friend is tutoring high school mathematics, and one of the techniques taught is to let an integral be $I$ then get $I = abc - I$ so that $I = abc/2.$ For example, $$
I := \int e^x\cos{x} dx = eˣ \...
- 473
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0
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38
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Divergence Test [closed]
The divergence test is inconclusive for $f(x) = 1/x$. The sum of the $1/n$'s is in fact divergent, $n\in \mathbb{N}$ and $n$ in $[1, ∞)$. We can say the same for the function $g(x) = 1/x^p$, with $0 &...
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1
answer
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Number of real roots of the n-th iteration of $f(x) = x^3 - 3x + 1$
Given, $$f(x) = x^3 - 3x + 1$$
I was solving a problem to find the number of distinct real roots of the composite function $f(f(x)) = 0$.
By analyzing the graph of $f(x)$, we can observe the local ...
- 964
3
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0
answers
96
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Showing $\int_0^a\left(f(x)-\frac12x\right)^2dx\leq\frac1{12}a^3$ for $f(x)\geq0$ satisfying $\left(\int_0^tf(x)dx\right)^2\geq\int_0^tf^3(x)dx$ [duplicate]
Problem:
Given positive value $a$, we have $f(x) \geq 0,\forall x\in[0, a]$, and
$$\left(\int_0^t f(x) dx\right)^2 \geq \int_0^t f^3(x)dx, \quad\forall t \in [0, a]$$
Show that
$$\int_0^a \left(f(x)-\...
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132
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Maximizing a function with infinitely many parameters
If I had a function $f$ that was this:
$$f\left(a_{0},a_{1},a_{2},a_{3},...a_{\infty},b_{0},b_{1},b_{2},b_{3},...b_{\infty}\right)=\frac{\sum_{k=0}^{\infty}\frac{1}{k+1}\left(\sum_{l=0}^{k}b_{l}\left(...
user1707124
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0
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What is the relationship between the two different definitions of Concave Function? [duplicate]
Some articles indicate the definition of a concave function $f(x)$ as follows:
$$\forall x_1,x_2\in D_f, \forall\lambda\in(0,1): f\left((1-\lambda)x_1+\lambda x_2\right) > (1-\lambda)f(x_1) + \...
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1
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334
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What is $\lim_n \sqrt[n]{1+\cos(n)}$?
As I was going through some exercise list with limits, I found $\lim_n \sqrt[n]{1+\cos^2(n)}$. This is easy enough, since $\cos^2$ is bounded between 0 and 1, so a squeeze theorem argument lets us ...
- 13.2k
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1
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43
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Proof of average rate of change of a function
I was learning about derivatives and I saw that the slope of the secant line between two points is the average rate of change of the function between the two points. But, the average, as we normally ...
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5
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1
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286
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The Multiplicative Role of $dx$ in Indefinite and Definite Integrals: A Comparison with Derivative Notation
I understand from prior discussions (e.g., What does the $dx$ mean in the notation for the indefinite integral?) that $dx$ in $\int f(x) \, dx$ serves as more than mere notation for the variable of ...
- 321
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2
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116
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Problem with solving $U_t=U_{xx}+\sum_{n=1}^\infty \frac{e^{-nt}}{n!} \sin (nx)$
Here is the problem:
Solve the following P.D.E with the given initial and boundary conditions:
$$U_t=U_{xx}+\sum_{n=1}^\infty \frac{e^{-nt}}{n!} \sin (nx), \quad U(0,t)=U(\pi,t)=U(x,0)=0$$
So, I ...
- 8,489
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81
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Evaluate $\lim_{n\to\infty}\left(\prod_{k=1}^n\left(1+\frac{k}{n}\right)\right)^{1/n}$ [duplicate]
I am trying to evaluate the limit
$$
\lim_{n\to\infty}\left(\prod_{k=1}^n\left(1+\frac{k}{n}\right)\right)^{1/n}.
$$
My first thought was to take logarithms. That turns the product into a sum and ...
- 19
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0
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Why does $\lim_{x \to 0}a_n = -C_n(g(1/x))$?
Define the sequence of functions $a_n$ such that $a_0 = f(x)$ and $a_n = \int a_{n-1}dx$ if $n \geq 1$ and $f(x)$ is an analytic function. Then $\lim_{x \to 0}a_n = -C_n(g(1/x))$ for all $n \geq 0$ if ...
- 127
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2
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75
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Having problem understanding the answer to the $1$D Heat Equation with the initial condition $U(x,0)=u_0-u_0\sin x$
Here is a problem and answer from my notes, where I have issues understanding parts of the provided answer.
Problem:
Solve the P.D.E. $U_t=U_{xx}$ with the following initial and boundary conditions:
$...
- 8,489
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100
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Is this method of deriving n!<0 using finite differences valid
From finite differences of $x^n$
I have
$D_nx^n=n!$
which $\implies (D_nx^n)/n!=1$
and telescoping this gives
$S_n=1+1+1+\ldots+1+1+0$, i.e., $= n$ and $D_{n+1}=0$
I have included the integer 1 to ...
- 19
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votes
1
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When is it justified to take a limit inside a series? [closed]
Let
$$
f(x)=\sum_{k=1}^{\infty}(-1)^k(k+1)\,\chi_{\left(\frac1{k+1},\,\frac1k\right]}(x),
\qquad x\in(0,1].
$$
Thus $f$ is constant on each interval $\left(\frac{1}{k+1},\frac{1}{k}\right]$, taking ...
- 537
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2
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Find $\sum\limits_{i,j,n\ge1}\frac{n + j + i}{n j i (n + j)(n + i)(j + i)}$
how to find the following series:
$$\sum_{i,j,n\ge1}\frac{n + j + i}{n j i (n + j)(n + i)(j + i)}$$
what i attempted was using symmetry like this
\begin{align*}
\sum_{i,j,n \ge 1} \frac{n + j + i}{n j ...
- 83
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0
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75
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Does the integral $\int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)}\,dx$ have a known closed form? [duplicate]
I am studying the definite integral
$$
I = \int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)}\,dx .
$$
The integral does converge:
as $x \to 0$, $\ln(1+x) \sim x$ and $\ln(1-x) \sim -x$, so the ratio tends to $-...
1
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1
answer
98
views
Must $g(4)$ satisfy any inequality if only $g(1)$, $g'(1)$, and $g''(1)$ are known?
Consider an everywhere differentiable function $g(x)$. Suppose we are given only the following information at the single point $x=1$:
$g(1) = 7$, $g'(1) = -3$, $g''(1)>0.5$
We are asked which ...
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Convergence of $\sum_{n=1}^{\infty} a_n,$ $\text{ where }$ $a_n = \prod_{k=1}^{n} \sin^2(2^k x)$ $\text{ and }$ $x \in (-\infty, +\infty)$ [closed]
To determine the convergence of the series $$\sum_{n=1}^{\infty} a_n, \text{ where } a_n = \sin^2 x \sin^2 2x \dots \sin^2 2^{n}x \text{ and } x \in (-\infty, +\infty).$$
I attempted to use the ratio ...
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0
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107
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How to extend series coefficient for negative coefficients?
Assume that $f(x)$ is such that there is a $g(x)$ such that $C_k(g(x)) = k!C_k(f(x))$ for all $k \geq 0$. It follows that $$\lim_{x \to 0}{\underbrace{\int\ldots\int}_{k \text{ times}} f(x)dx \ldots ...
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9
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Evaluate $\int_0^{\frac\pi2}\frac{\mathrm dx}{a\sin ^2x+b\cos ^2x}$ without using its antiderivative
Find the value of
$$\int_0^{\frac\pi2} \frac{1}{a \sin ^2x+b \cos ^2x} \, \mathrm dx,$$
where $a$, $b>0$.
The corresponding indefinite integral evaluates to
$$\int \frac{1}{a \sin ^2x+b \cos ^2x} \...
- 5,070
3
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1
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91
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$f(0)=1$, $f(x) \ge 0 \ge f'(x)$, $f''(x)\le f(x)$ for $x\ge 0$
Problem
Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0$ and $f''(x) \le f(x)$, for all $x\ge 0$. Show ...
- 3,478
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1
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Range of base $a$ such that $f(x) = a^x - bx + e^2$ has two distinct zeros for all $b > 2e^2$
I am trying to solve the following problem involving a function with parameters $a$ and $b$.
The Problem:
Given the function $f(x) = a^x - bx + e^2$ where $a > 1$ and $x \in \mathbb{R}$.
Discuss ...
0
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2
answers
100
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Absolute convergence and rearrangements
I have a few questions regarding the author’s proof of the following theorem:
I don't understand the part where the author claims that absolute value of each term of $t_m - s_N$ is in the tail of the ...
- 151
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1
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how to solve $\mathcal{J}=\int_0^{\pi/2} \frac{x \arcsin(\cos x)}{\sqrt{1 + \sin^2 x} + \cos x} \, dx$
what I tried was that $x \in [0, \pi/2]$ meaning $\cos x \in [0,1]$. therefore
$$
\arcsin(\cos x) = \frac{\pi}{2} - x
$$
so the integral is
$$
\mathcal{J} = \int_{0}^{\pi/2} \frac{x\left(\frac{\pi}{2} ...
- 71
3
votes
2
answers
96
views
Problem with understanding the technique used to calculate $B_n=\frac{2u_0}{\pi}\int_{x=0}^\pi\sin(nx)(\sin x+\sin 3x)dx$ for heat equation.
This is a problem and answer from my notes:
Solve heat equation for $l=\pi$ and with the initial and boundary conditions:
$U(0,t)=U(\pi,t)=0;\;\;U(x,0)=u_0(\sin x+\sin 3x)$
The answer to the above ...
- 8,489
-3
votes
0
answers
49
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Divergence of a series based on the sequence [closed]
If a sequence [an] does not converge, that is the limit of "an" as n tends to infinity" does not exist, will the series be referred to as convergent or divergent??
-1
votes
2
answers
61
views
Does uniformly continuous functions apply to something like "sandwich theorem"? [closed]
Suppose $f,g$ are two uniformly continuous functions on $\mathbb R$, and $h$ is a continuous function on $\mathbb R$ that satisfies:$$f(x)\le h(x) \le g(x)$$Does that mean $h$ is a uniformly ...
- 9
2
votes
1
answer
123
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Can we prove that the limit $e = \lim_{h \to 0^+} (h + 1)^{\frac1h}$ exists using the fact that the function in the limit always decreases for $h>0$?
(If you want to see the specific step that I just want to make sure is valid, it's the one at the very bottom; there is a comment in parentheses that goes before it.) I was trying to prove that the ...
- 315
0
votes
0
answers
83
views
Limit of the function satisfying $f(x)=x-f(x^2)$ as $x\to 1^-$
I guess $\lim\limits_{x\to 1^-} f(x) = 1/2$, where the function $f(x)$ defined by $f(x)=x-f(x^2)$ in $[0,1)$, or by the series:
$$
f(x) = x - x^2 + x^4 - x^8 + x^{16} - x^{32} + \cdots.
$$
I know $f(x)...
- 81
1
vote
0
answers
105
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Equivalent definitions of vector-valued Riemann integral
Let $X$ be a Banach space and $[a, b]$ a finite nondegenerate closed interval. I wish to define the Riemann integral of a function $f : [a, b] \to X$. There are two natural definitions that I think ...
- 7,769
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0
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95
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+100
Practical and historical role of Jordan measure [migrated]
In my earlier questions, the proofs given by Asigan and D.R. showed that the Jordan outer/inner measure of the subgraph $[0,f]$ and the Darboux upper/lower integrals of $f$ are essentially the same ...
- 4,220
2
votes
1
answer
135
views
Prove for all $n \in \mathbb{N}, \int_{0}^{1}(f(x))^{n+1}dx \geq \int_{0}^{1} x^n f(x)dx$
Let $f: [0,1] \rightarrow \mathbb{R}$ a continuous function such that $f(x) \geq 0$ for all $x \in [0,1]$ and $$\int_{x}^1 f(t)dt \geq \frac{1-x^2}{2} \text{ for all } x \in [0,1].$$
Prove for all $n \...
- 359
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votes
1
answer
74
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Is this the correct formula for parameterizing a Torus?
I am going over my professor's lecture notes, and I'm pretty sure he made a mistake. He has made errors in the past, so I just want to confirm if this equation is correct or if I am mistaken.
He has ...
4
votes
1
answer
217
views
Real variable method to show that $\int_{-\infty}^\infty \frac{\sinh ax}{\sinh \pi x}\cos bx dx = \frac{\sin a}{\cos a+\cosh b}$?
There are evaluations of the above integral using complex analytic techniques, but is there a real variables way to show it? I've tried the "Feynman differentiation under the integral trick" ...
- 140
0
votes
2
answers
147
views
Evaluate (when possible) $\sum_{n=0}^{+\infty}\ln\left(2\cos\frac{\alpha}{2^n}-1\right)$
This appeared on the exercises sheet for a "Numerical Series" chapter of a university course: "Determine the nature and the possible sum of the numerical series". Among 18 examples ...
- 197
0
votes
0
answers
39
views
Confusion when projecting a function onto a function basis
Let $\phi_i(x)$ denote the Fourier basis vectors on $[-1, 1]$. Such that if $i$ is even it's a cosine and if it is odd it is a sine.
$\phi_0(x)$ is just the constant 1.
So its standard norm is 2 on ...
- 3,857
0
votes
0
answers
70
views
Question about integral of a multivariable function
How should I simplify this expression?
$$g'(t)\cdot \int f(x)\,dx$$
Where $t$ is a constant relative to $x$.
I have a few ideas for what it might be, but I’m new to integrals of functions with ...
- 382
0
votes
1
answer
87
views
How do I differentiate for a variable ($t$) a definite integral of another variable ($x$)? [closed]
I was reading a pdf for Feynman's trick for integration, and at some point this function is defined:
$f(t):=\int_{0}^{1} \frac{x^{t}+1}{\log(x)}dx$
and later it says:
$f'(t)=\int_{0}^{1} x^{t}dx=\frac{...
- 1