Question

Showing $ \int_{\mathbb{R}^{2}_{+}} dt_1 dt_2 e^{-[t_1^2 + 2(-y-i0) t_1 t_2 + t_2^2]}= \frac{1}{2} \int_0^{\infty} \frac{dt}{t^2+2(-y-i0)t+1} $

score 43 · Answer 1 · 2026-04-16 22:55:52

Showing $ \int_{\mathbb{R}^{2}_{+}} dt_1 dt_2 e^{-[t_1^2 + 2(-y-i0) t_1 t_2 + t_2^2]}= \frac{1}{2} \int_0^{\infty} \frac{dt}{t^2+2(-y-i0)t+1} $

score 43 · Answer 2 · 2026-04-17 07:48:12

43

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List Question

Showing $ \int_{\mathbb{R}^{2}_{+}} dt_1 dt_2 e^{-[t_1^2 + 2(-y-i0) t_1 t_2 + t_2^2]}= \frac{1}{2} \int_0^{\infty} \frac{dt}{t^2+2(-y-i0)t+1} $

Evaluating $\int_{\mathbb{R}^2_{+}} \frac{ds_1 ds_2 s_2^{\epsilon} e^{- \frac{s_1 s_2 q^2}{s_1+s_2}}}{(s_1+s_2)^{1-\epsilon}}$

Solving an integral that comes up in the 3D screened Poisson equation

On the Divergence of an Improper Integral where the Integrand becomes unbounded in the Neighbourhood of a Point inside the Interval of Integration

How to evaluate $\int_0^1\ln(1-x)dx$

Problem with convergence of integrals

Does bounded area of a curve imply bounded volume of associated solid of rotation?

Showing the convergence of improper integral

Proving $\int_0^\infty t e^{2t} e^{-e^{-2t}} \mathrm{d}t = \frac{-\gamma}{4}$.

Improper integral with singular integrand in three dimensions

A specific "gaussian" integral

Applying Fubini's theorem to find the expected value of a random variable

Show the integral equality $\int_0^\infty\frac{1}{\sqrt{x^3+1}}\mathrm dx = 2 \int_0^\infty\frac{1}{\sqrt{x^6+1}}\mathrm dx$ without evaluating

The integral $I(a,b)=\int_0^1x^a\log^b(x) \space dx$

How to prove this integral is convergent

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