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Apr
27
revised On the solution of Volterra integral equation
added 181 characters in body
Apr
27
answered On the solution of Volterra integral equation
Apr
27
comment On the solution of Volterra integral equation
Where is your Volterra operator defined? On which algebra of functions are you seeking a solution to this integral equation?
Apr
19
comment criteria of convergence of a series
Use the integral test followed by the change of variable $x = \ln(t)$
Apr
17
answered Does $\int _1 ^\infty\frac {f(x)} x\,dx$ converge or diverge?
Apr
17
comment If $f$ is entire and $\lim_{z\to\infty} \frac{f(z)}{z} = 0$ show that $f$ is constant
Yet another method might be to consider the function $g(z) := f(\frac{1}{z})$ and prove that g is entire (only have to check that g has a removable singularity at 0). Now go back to the Taylor expansion of f at $\infty$ and conclude
Apr
9
comment What does it mean “a Lebesgue point of $f$”?
It is the same thing, we just expressed it in terms of the definition of a limit. Notice that $L^{p}_{loc} \subset L^{1}_{loc}$ for $p\geq 1$
Apr
6
comment Prove $\sum_{k=0}^{n}{n \choose k}(-1)^k \frac{1}{k+1} = \frac{1}{n+1} $
Interchange the integral with the sum and use the binomial theorem
Mar
26
comment Old & cool integral $\int_0^{\pi} \sin^{b-1}(x) \sin(a x) \ dx=\frac{\pi \sin(a \pi/2)}{2^{b-1}b B\left(\frac{b+a+1}{2},\frac{b-a+1}{2}\right)}$
Have you tried using the Fourier series expansion of $\sin(ax)$ ?
Feb
4
comment Computation of an iterated integral
Very good Answer! (+1)
Feb
2
comment If $f(x)$ is continuous at $ x=0$
Your assumtion is that $f = O(x^{2})$ near $x=0$
Jan
30
answered Reflexive Banach spaces, compactness
Jan
30
answered Problem calcualting Fourier coeff. of tent function.
Jan
22
comment Do these series converge uniformly?
Hints: For the first one, what can be said about the series for $x\leq 0$? For the second one, try splitting the sum
Jan
11
revised Integration of $\int_0^{+\infty} \frac{\ln(1/x)}{1-x^2}\,dx$
added 2 characters in body; added 88 characters in body
Jan
11
answered Integration of $\int_0^{+\infty} \frac{\ln(1/x)}{1-x^2}\,dx$
Jan
10
comment Deciding whether a function $f$, defined on $\mathbb{R}^2$, is Lebesgue-integrable over two sets
For (ii): If $f(x,y)$ where integrable, then the integral of $|f(x,y)|$ over $A$ should be finite, independent of the order of integration. But..
Jan
10
comment Prove that $τ(m^n)$ and $n$ are coprime $(m,n ∈ N^+)$
$m^{n}$ might have more distinct divisors, if say $m$ is not a prime. But there is a way to work around it, by utilizing the multiplicativity of $\tau$
Jan
7
revised On uniqueness of extension of linear functional on $\ell^{1}$
added 29 characters in body
Jan
5
comment Show that $\int_0^1 f(x)g(x)\,dx=0$ for all $g\in C([0,1])$ implies $f=0$ almost everywhere
Well, that works as long as we remain continuous.