Tagged Questions
4
votes
1answer
53 views
The geometric interpretation [duplicate]
In the course of mathematical analysis, there was one problem that i excited to know more about it:
What is the geometric interpretation of
$$ \int_a^b f(x)\,d(\alpha(x)) $$
and $\alpha(x)$ is ...
5
votes
1answer
89 views
A problematic integral: $\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$
Is there a special trick to calculate this integral?
$$\int_0^{2\pi} e^{-2\pi i\lambda\cos(t)}\,dt$$
for $\lambda>0$.
1
vote
0answers
29 views
Show derivative of integral equals integral of partial derivative if M[0,1]-measurable
I am trying to determine a method of approaching the following:
Suppose that $f:[0,1] \times (0,1)$ $\rightarrow$ $\mathbb{R}$ is such that, for each $y \in (0,1)$, the function $f^{[y]}(x) = f(x,y)$ ...
1
vote
2answers
69 views
Function that is Riemann-Stieltjes integrable but not Riemann integrable?
This is my first question, so please go easy on me :3 - I've searched, and I haven't found any questions that are particularly similar to this one.
I'm reading Rudin's Principles of Mathematical ...
3
votes
1answer
56 views
Highly Oscillating Integrals
I'd like to know the behavior of integrals of the form:
$$
\int_0^1 f(x) \cos(k x) dx
$$
as $ k \rightarrow \infty $ where f is a smooth function. It is easy to see, by expanding f in power series, ...
1
vote
1answer
57 views
Computing an explicit solution to an integral equation via the Neumann Series.
I am hoping that someone can provide guidance for solving the integral equation
$$u=f+\lambda Au$$
where $1/\lambda\notin\sigma(A)$, $f\in L^2[0,2\pi]$, and $A:L^2[0,2\pi]\to L^2[0,2\pi]$ is defined ...
1
vote
2answers
95 views
Integral with hyperbolic functions
I need to compute:
$$ \int_{x^2+y^2=1} \frac{\sinh(x)dy- \sin(y)dx}{\cosh(x)-\cos(y)}$$ where the circle $x^2 + y^2 = 1$ is oriented anticlockwise. So, can somebody show me how? I found the ...
0
votes
1answer
16 views
Calculate volume of inequality
$\{(x,y,z) \in \mathbb{R}^3 | 2*max(|x|,|y|)^2+z^2\leq 4\}$
any tips for me anyone?
i made a sketch but what now?
1
vote
1answer
65 views
Concept of integration to differential form
How to integrate differential form actually. As far as I know, a differential form is a multilinear function mapping from a vector space to a real number. Let's take $\int_c fdx+gdy$ as an example. It ...
0
votes
0answers
48 views
About calculating a Green's function
For a positive integer $d>=2$ and a real number $h_0$ I have the differential equation for a function $f$ of $x$,
$x^2 f'' + h_0^2 x f' = \frac{d(d-2)}{2} f$
The eigenfunctions are then $f \sim ...
1
vote
0answers
25 views
Bounds for the exponential integral
In Abramowitz and Stegun: Handbook of Mathematical Functions
(on page 229, property 5.1.20) it is found that
$$
\frac{1}{2} \log \left(1 + \frac{2}{x} \right) < \exp(x) E_1(x) < \log \left(1 + ...
0
votes
2answers
87 views
Find a real value $y$ for that the identity holds
Question: Find a real value $y$ such that the following identity holds
$$\log(\log(\log(x)))=\int^x_y\frac{dx}{x\log(x)\log(\log(x))}\quad (x>y)$$
Then, find an algorithm to calculate
...
2
votes
1answer
42 views
How to recover a measure from its Fourier transform?
Let $f$ be the complex function defined on $\mathbb{R}$ by
$$
f(t)=\frac{1-it}{1+it}.
$$
1) Does there exist a complex bounded measure $\mu \in M(\mathbb{R})$ such that $\hat{\mu}=f$ (where $\hat{}$ ...
0
votes
1answer
31 views
Fourier transform of $\mathrm{rec}(x) =\begin{cases} 1 & \text{if }|x| < 0.5,\\ 0.5& \text{if }|x| = 0.5,\\ 0& \text{if }|x| > 0.5 \end{cases}$
$$\mathrm{rec}(x) =\begin{cases}
1 & \text{if }|x| < 0.5,\\
0.5& \text{if }|x| = 0.5,\\
0& \text{if }|x| > 0.5
\end{cases}$$
The Fourier transform of this function is ...
1
vote
1answer
63 views
Show that this equation is true.
Consider the following function in $\mathbb{R}^n (n\geq 3)$:
\begin{equation}
H(y)=2b_n\int_{0}^{\infty}e^{\ as}D_n\Phi(y-\tilde{x}+bs)\text{ d} s,\quad (x, y\in\mathbb{R}_{+}^{n}, x\neq y),
...
1
vote
3answers
74 views
Change of variables in two dimensions
This is from Munkres' Analysis on Manifolds, Section 17, Question 4.
(a) Show that
$$ \int_\Bbb {R^2} e^{-(x^2+y^2)} = \left[ \int_\Bbb R e^{-x^2}\right]^2,$$
provided the first of these ...
1
vote
2answers
125 views
If $f(x) \le g(x) \le h(x)$ for all $x\in D$ and $f$, $h$ are Riemann integrable, then so is $g$. True or False? (Check my work) [duplicate]
If $f(x) \le g(x) \le h(x)$ for all $x\in [a,b]$, and f and h are Riemann integrable on
[a,b], then so is g. True or false? Explain.
A: True
Proof:
Since $f\in R[a,b]$, $\bar \int_a^b{f} = ...
0
votes
0answers
34 views
Taylor expansion of an integral in spherical co-ordinates
I've some difficulty deriving this equation from jackson electrodynamics (The equation after 1.30)
$\nabla^2 \Phi_a\left({\textbf{x}}\right)=-\frac{1}{\epsilon_0}\int_{0}^{R} ...
6
votes
4answers
172 views
why $\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$ means surface area?
Why the following integral means the area of surface $f(x,y)=z$?
$$\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$$
1
vote
3answers
75 views
How to determine whether an integral is convergent
I missed up the last lecture and can't understand how to determine whether an integral with parameters is convergent or divergent?
For example: For which values of the parameters $p,q \in ...
6
votes
1answer
65 views
Cauchy Integral Formula for Matrices
How do I evaluate the Cauchy Integral Formula $f(A)=\frac{1}{2\pi i}\int\limits_Cf(z)(zI-A)^{-1}dz$ for a matrix ...
0
votes
3answers
69 views
Uniform convergence of sequence of functions
Given that $$\gamma_n\rightarrow\gamma$$ uniformly, can we conclude that $$\int^b_a\|\gamma_n'\|\rightarrow\int^b_a\|\gamma'\|$$ uniformly?
I know that we even do not have ...
2
votes
1answer
38 views
how to find limit of $\lim_{n\to+\infty}\int_0^{+\infty}|f(x+\frac1n)-f(x)|dx$?
Assume $f$ is continuous on $\mathbb R$ and $\int_0^{+\infty}|f(x)|dx\lt+\infty$. Then how does one find $$\lim_{n\to+\infty}\int_0^{\infty}\left|f\!\left(x+\frac{1}{n}\right)-f(x)\right|dx?$$
Thanks ...
3
votes
2answers
222 views
$3\int_{0}^{1}(f'(x))^2dx \geq (2\int_{0}^{1}f(x)dx)^2 \impliedby 2\int_{0}^{\frac{1}{2}}f(x)\,\mathrm dx=\int_{\frac{1}{2}}^{1}f(x) \,\mathrm dx$
Let $f : \mathbb{R} \to \mathbb{R} $ be a differentiable function. Suppose that $2\int_{0}^{\frac{1}{2}}f(x)\,\mathrm dx=\int_{\frac{1}{2}}^{1}f(x) \,\mathrm dx$
Show that $$3\int_{0}^{1}(f'(x))^2 ...
2
votes
2answers
27 views
Problems with determining convergence of integral
It should be easy but I'm not sure... For which $\alpha \in \mathbb{R}$ the following integral is convergent:
$$\int_0^1 \int_0^1 \frac{1}{|y-x|^\alpha}dxdy \ \ ?$$
I get for all $\alpha \neq 1,2$ ...
2
votes
4answers
122 views
Gaussian-like integral : $\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$
Prove that :
$$ \frac{\sqrt{\pi}}{2} e^{-\frac{a^2}{4} } =\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$$
the only thing I can think of is differentiating the RHS and trying to get :
$$ -2 ...
2
votes
2answers
41 views
Integrating factor of a differential arising from thermodynamics
Let $\delta E = (xy^2 + xye^x)dx + (2x^2y + xe^x)dy$
I now need to find the integrating factor $\mu (x,y)$ s.t. $dS = \mu (x,y) \delta E$ is a exact differential.
Now as far as I know $\delta E$ is ...
0
votes
1answer
75 views
Proof of integration-by-substitution (two questions)
Here's a version of the theorem:
$$\int_a^b f(g(x))g'(x)dx=\int_{g(a)}^{g(b)} f(u)du$$ provided that:
$f$ is continuous on an interval $I$,
$g'$ is continuous on $[a,b]$,
$g[a,b]=I$ ...
2
votes
3answers
130 views
Prove integral is greater than $0$
$f(x)$ is Riemann integrable on $I=[a,b]$ and $f(x)>0$ for all $x \in I$, prove $\int_a^b f(x) dx >0$ .
Need help on this question, please help me
5
votes
2answers
178 views
Integral $\int_0^{\infty} \sin(x^2)/x^2\,dx$
Does anyone have a proof for $$\int_0^{\infty}\frac{\sin(x^2)}{x^2}\,dx=\sqrt{\frac{\pi}{2}}.$$
I tried to get it from contour integrating $$\frac{e^{iz^2}-1}{z^2},$$ but failed.
Thanks.
1
vote
1answer
61 views
Proof of $\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$?
Can anyone show a proof of $$\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$$
where $0<a<1$, and $$\Gamma(a)=\int_0^\infty t^{a-1}e^{-t}\,dt.$$ Thank you.
1
vote
3answers
65 views
Evaluating a complex integral over a half-ring
I need to integrate the $z/\bar z$ (where $\bar z$ is the conjugate of $z$) counterclockwise in the upper half ($y>0$) of a donut-shaped ring. The outer circle is $|z|=4$ and the inner circle is ...
1
vote
1answer
73 views
how compute $\int_a^b xf(x)dx$? such that$ f(a+b-x)=f(x)$
let $f(a+b-x)=f(x)$ then how compute $$\int_a^b xf(x)\,dx$$ thanks for any hints
3
votes
0answers
129 views
Under which hypotheses is switching between sum and integral signs legit?
Which hypotheses are needed to change the order of sum and integral signs?
Concrete example: consider the expression
$$
...
3
votes
2answers
53 views
If $g\in C^1([0,1],\mathbb{R})$, show $\lim_{x\to+\infty }\int_0^1x^ndg(x)=0$
Suppose $g:[0,1]\to\mathbb R$ such that $g\in C^1$ (i.e., $g'$ exists and is continuous). I want to show $$\lim_{n\to+\infty }\int_0^1x^ndg(x)=0.$$ Thanks.
0
votes
1answer
61 views
Special function or Weierstrass?
Given that $f(x)$ is integrable on any $[0,a]$, such that $\lim_{x\to+\infty} f(x)=1$
Prove that: $$\lim_{t\to 0+}\int_0^{\infty}t\text{e}^{-tx}f(x)\text{d}x=1$$
1
vote
2answers
50 views
Evaluating the integral $\int\int_G \frac{\ln{(x^2+y^2)}}{x^2+y^2} dx dy$
I need to evaluate the following integral:
$$\int\int_G \frac{\ln{(x^2+y^2)}}{x^2+y^2} dx dy$$
Here $G=\{(x,y)\in \mathbb{R}^2: 1 \leq x^2+y^2 \leq e^2\}$. I tried to use parametric transformations ...
4
votes
0answers
109 views
Integral of a gaussian function of trigonometric functions
I need help with the analytical solution of this integral:
...
6
votes
2answers
86 views
A problem of evaluating an integral
How to prove that
$$\int_{0}^{1}(1+x^n)^{-1-\frac{1}{n}}dx=2^{-\frac{1}{n}}$$
I have tried letting $t=x^n$,and then convert it into a beta function, but I failed. Is there any hints or solutions? ...
0
votes
0answers
28 views
Existence of Young-Stieltjes Integral
Can anybody help me about the existence of Young-Stieltjes Integral?
I'm confused about how to proof the existence of Young-Stieltjes Integral :(
0
votes
0answers
39 views
Young-Stieltjes Integral
Can anybody explain to me about how to construct a Young-Stieltjes Integral from Riemann-Stieltjes Integral? I have a problem when i'm reading the paper from Young :(
Here is the theorem about ...
6
votes
1answer
106 views
how prove that $|f(x)|\ge2^n (n+1) $ under these conditions?
Assume $f:[0,1]\mapsto\mathbb{R}$ is continuous and satisfies
$\int_0^1x^kf(x) \, dx=0 \quad\forall k\in\{0,1,2,\ldots,n-1\}$,
$\int_0^1x^n f(x) \, dx=1$.
How do you prove that $\exists x\in[0 ...
4
votes
3answers
66 views
For all integrable $f:[-1,1]\mapsto \mathbb{R}$ prove that $\int_{-1}^1f^2(x)\ge\frac12(\int_{-1}^1f(x))^2+\frac32(\int_{-1}^1xf(x))^2$
For all integrable $f:[-1.1]\mapsto \mathbb{R}$ peove that $$\int_{-1}^1f^2(x)dx\ge\frac12\left(\int_{-1}^1f(x)dx\right)^2+\frac32\left(\int_{-1}^1xf(x)dx\right)^2$$
Thanks in advance.
1
vote
1answer
123 views
how prove this integral inequality?
How prove that for all continuous and decreasing function $f:[0 ,1]\mapsto(0,+\infty)$ $$\frac{\int_{0}^1x(f(x))^2dx}{\int_{0}^1xf(x)dx}\leq \frac{\int_{0}^1(f(x))^2dx}{\int_{0}^1f(x)dx}$$
thanks in ...
3
votes
0answers
65 views
Simplifying an integral arising in Physical Chemistry
I am struggling to understand the following transition (encountered in a paper on Physical Chemistry).
Let
$$D=\frac{\tau_0^{-1}\int_0^\infty G(t)dt}{1-\tau_0^{-1}\int_0^\infty G(t)\int ...
0
votes
2answers
131 views
Prove that there is a $\delta$ such that $\int_{0}^{1} (f(x))^2dx\leq \delta$$\int_{0}^{1} (f'(x))^2dx$ for all $f$ with these conditions
Let $S=\{f:\mathbb{R} \to \mathbb{R}\}$ that satisfies:
$\forall f\in S$, $f'$ exists and $f'$ is continuous
and $f(0)=f(1)=0$.
Please prove that $\exists \delta :\forall f\in S$ s.t.
$\int_{0}^{1} ...
2
votes
2answers
71 views
Showing that $\lim_{\delta \to 0^+} \frac{1}{\delta} \int_x^{x + \delta} f(t) \ \mathrm{d}t = f(x)$
I'm working on proving the following equation:
$\lim_{\delta \to 0^+} \frac{1}{\delta} \int_x^{x + \delta} f(t) \ \mathrm{d}t = f(x)$,
where $f$ is given to be Riemann integrable and continuous on ...
2
votes
2answers
74 views
Proof of Integral and Limes
I need an idea for the following statement:
$f: [a,b]\to\mathbb{R}$ continuous and $x\in (a,b)$ $\implies$ $\lim\limits_{h \rightarrow 0}{\frac{1}{2h} \int_{x-h}^{x+h} f(y) \, dy = f(x) }$
I just ...
1
vote
2answers
83 views
Simplifying an integral by introducing an additional parameter.
This is one of my homework tasks this week.
Calculate the integral $$I = \int_0^\infty dx\ x^3 e^{-x}$$ by introducing an additional parameter $\lambda$ and rewriting the exponential function as ...
4
votes
0answers
51 views
Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''dx=\int_0^ty'y''dx$.
Let $y(x)$ be a solution to $y''+e^xy=0$. Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''dx=\int_0^ty'y''dx$.
