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'list of improper integrals'
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| Title of object: |
list of improper integrals |
| Canonical Name: |
ListOfImproperIntegrals |
| Type: |
Topic |
| Created on: |
2009-01-10 08:31:04 |
| Modified on: |
2009-01-23 18:49:27 |
| Classification: |
msc:40A10 |
| Keywords: |
improper integral |
Revision comment (for changes between this and next version):
Preamble:
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% it should be fine as is for beginners.
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
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%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
\PMlinkid{1.}{7065} \; $\displaystyle\int_0^\infty e^{-x^2}\,dx \;=\; \frac{\sqrt{\pi}}{2}$\\
\PMlinkid{2.}{11495} \; $\displaystyle\int_0^\infty e^{-x^2}\cos{kx}\,dx\;=\;\frac{\sqrt{\pi}}{2}e^{-\frac{1}{4}k^2}$\\
\PMlinkid{3.}{11504} \; $\displaystyle\int_0^\infty \frac{e^{-x^2}}{a^2\!+\!x^2}\,dx
\;=\;\frac{\pi}{2a}e^{a^2}\,{\rm erfc}\,a$\\
\PMlinkid{4.}{10980} \; $\displaystyle\int_0^\infty\sin{x^2}\,dx \;=\; \int_0^\infty\cos{x^2}\,dx \;=\;
\frac{\sqrt{2\pi}}{4}$\\
\PMlinkid{5.}{7082} \; $\displaystyle\int_0^\infty\frac{\sin{x}}{x}\,dx \;=\; \frac{\pi}{2}$\\
\PMlinkid{6.}{11487} \; $\displaystyle\int_0^\infty\left(\frac{\sin{x}}{x}\right)^2 dx \;=\; \frac{\pi}{2}$\\
\PMlinkid{7.}{11487} \; $\displaystyle\int_0^\infty\frac{1-\cos{kx}}{x^2}\,dx \;=\; \frac{\pi k}{2}$\\
\PMlinkid{8.}{11480} \; $\displaystyle\int_0^\infty\frac{x^{-k}}{x\!+\!1}\,dx \;=\; \frac{\pi}{\sin{\pi k}}
\quad (0 < k < 1)$\\
\PMlinkid{9.}{11544} \; $\displaystyle\int_{-\infty}^\infty\frac{e^{kx}}{1\!+\!e^x}\,dx \;=\; \frac{\pi}{\sin{\pi k}}
\quad (0 < k < 1)$\\
\PMlinkid{10.}{7136} \; $\displaystyle\int_0^\infty\frac{\cos{kx}}{x^2\!+\!1}\,dx \;=\; \frac{\pi}{2e^k}$\\
\PMlinkid{11.}{11489} \; $\displaystyle\int_0^\infty\frac{a\cos{x}}{x^2\!+\!a^2}\,dx
\;=\; \int_0^\infty\frac{x\sin{x}}{x^2\!+\!a^2}\,dx \;=\; \frac{\pi}{2e^a} \quad\; (a > 0)$\\
12. \; $\displaystyle\int_0^\infty\frac{\sin{ax}}{x(x^2\!+\!1)}\,dx \;=\; \frac{\pi}{2}(1-e^{-a}) \quad\; (a > 0)$\\
\PMlinkid{13.}{9223} \; $\displaystyle\int_0^\infty e^{-x}x^{-\frac{3}{2}}\,dx \;=\; \sqrt{\pi}$\\
\PMlinkid{14.}{10637} \; $\displaystyle\int_0^\infty e^{-x}x^3\sin{x}\,dx \;=\; 0$\\
\PMlinkid{15.}{7891} \; $\displaystyle\int_0^\infty\!\left(\frac{1}{e^x-1}-\frac{1}{xe^x}\right) dx \;=\; \gamma$\\
\PMlinkid{16.}{11516} \; $\displaystyle\int_0^\infty\!\frac{\cos{ax^2}-\cos{ax}}{x} dx \;=\; \frac{\gamma+\ln{a}}{2} \quad (a > 0)$\\
\PMlinkid{17.}{11511} \; $\displaystyle\int_0^\infty\frac{e^{-ax}\!-\!e^{-bx}}{x}\,dx \;=\; \ln\frac{b}{a} \quad (a > 0,\;\, b > 0)$\\
\PMlinkid{18.}{11526} \; $\displaystyle\int_1^\infty\left(\arcsin\frac{1}{x}-\frac{1}{x}\right)\,dx \;=\; 1+\ln{2}-\frac{\pi}{2}$\\
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