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Questions tagged [singular-integrals]

Singular integrals are integral operators with kernel $K:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ that is not defined on the diagonal $x=y.$ This tag is for questions related to singular integrals and their applications.

A singular integral is an integral operator of the form

\begin{equation*} Tf=\int K(x,y)f(y)dy \end{equation*}

defined for "nice" (ie, Schwartz) functions and with kernel $K$ possessing a singularity at the origin. They can be of convolution type (the kernel must be locally integrable on $\mathbb{R}^n$ without the origin) or non-convolution type.

The most basic examples are the Hilbert transform, and its generalisation, the Riesz transform.

169 questions
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Singular expansion of an integral

I consider the following function on $\mathbb{R}$ : $$ f(z)=\int\limits_0^{+\infty} dx\frac{1+x}{\sqrt{x}}\frac{1}{x\, e^x +z}$$ I would like to know its singular expansion at $z=0$. Doing a series expansion of the denominator does not help me much,…
Alexandre Krajenbrink
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Integrable function on positive axis.

Suppose that $f(x)$ is continuous on $(0,\infty)$ and that $$\int_0^{\infty} \frac{f(x)}{x} \,dx < \infty.$$ Does it follow that $$\int_0^{\infty}\frac{f(x)^2}{x} \,dx < \infty ?$$
user385459
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Why $\int_0^{1/2}\frac{dx}{(x\ln^2x)^{1+\epsilon}}$ is infinite for $\epsilon > 0$

Using substitution ($u=\ln x$), it can be shown that $\int_0^{1/2}\frac{dx}{(x\ln^2x)}$ = $\frac{1}{\ln2}$. However, if we raise the integrand to a number slightly greater than one, this integral diverges, and we're having trouble showing this.…
letslearnmath
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