\(\S\S\) 7.6 Q[5] \(\int_0^\infty \frac{x^p}{(x^2+a^2)^2}\, dx\)
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Notices
Complex Variables --- Principles and Problem Sessions :: Start-Page
- Complex Variables --- Principles and Problem Sessions === SOLUTIONS and ERRATA
- Solutions to Problem Sessions
- \(\S\S 3.6\) :: Discontinuity across the branch cut
- Chapter 7 :: Contour Integration
- \(\S\S\) 7.1 Tutorial \( \int_0^\infty Q(x)\, dx \)
- \(\S\S 7.2\) Tutorial :: Improper Integrals of Rational Functions
- \(\S\S\) 7.3 Exercise ----- Integrals of Type \(\int Q(x) \,dx\)
- \(\S\S\) 7.3 Q[1] \(\int_0^\infty\frac{1}{(x^2+p^2)^2} dx \)
- \(\S\S\) 7.3 Q[2] \(\int_0^\infty \frac{x}{x^4+1} dx\)
- \(\S\S\) 7.3 Q[3] \(\int \frac{dx}{x^4+x^2+1}\)
- \(\S\S\) 7.3 Q[4] \(\int_{-\infty}^\infty \frac{dx}{(x^2+x+1)^2}\)
- \(\S\S\) 7.3 Q[5] \int_{-\infty}^{\infty}\(\frac{(x^2+2)}{(x^4+10x^2+9)}\)
- \(\S\S\) 7.3 Q[6] \(\int_0^\infty \frac{(x-2)}{(x^4+10x^2+9)} dx\)
- \(\S\S\) 7.3 Q [7] \(\int_0^\infty \frac{x^2\, dx}{x^6+1}\)
- \(\S\S\) 7.3 Q [8] \(\int_0^\infty \frac{dx}{(x^2+p^2)(x^2+q^2)}\)
- \(\S\S\) 7.3 Q[9] \(\int_0^\infty \frac{x^2}{x^4+1}dx \)
- \(\S\S\) 7.3 Q[10] \(\int_0^\infty \frac{x^2}{x^{2N+1}+1}\, dx\)
- \(\S\S\) 7.3 Q[11] \(\int_0^\infty \frac{x^4}{x^{2N}+1} dx\)
- \(\S\S\) 7.3 Q[12] \( \int_0^\infty \frac{dx}{x^N+1} \)
- \(\S\S\) 7.3 Q[13] \(\int_0^\infty \frac{x^{2N}}{x^{2M}+1} \, dx\)
- \(\S\S\) 7.3 Q[14] \(\int_0^\infty \frac{dx}{(p+qx^2)^N}\)
- \(\S\S\) 7.4 Exercise ------ Integrals of \({\sin x \choose \cos x}\) with Rational Functions
- \(\S\S\) 7.4 Q[1] \(\int_0^\infty \frac{x\sin p x\, dx}{x^2+1} \)
- \(\S\S\) 7.4 Q[2] \(\int \frac{\cos p x\, dx}{(x^2+1)^2}\)
- \(\S\S\) 7.4 Q[3] \(\int_0^\infty \frac{\cos px\, dx}{(x^2+1)(x^2+4)}\)
- \(\S\S\) 7.4 Q[4] \(\int_0^\infty \frac{\sin x \, dx}{(x^2+4)(x^2+9)}\)
- \(\S\S\) 7.4 Q[5] \(\int_0^\infty \frac{x^2\cos px \, dx }{x^4+1}\)
- \(\S\S\) 7.4 Q[6] \( \int_0^\infty \frac{x\sin p x\, dx}{x^4+1}\)
- \(\S\S\) 7.4 Q[7] \(\int_0^\infty \frac{x\cos \alpha x\, dx}{x^2-2x+10} \)
- \(\S\S\) 7.4 Q[9] \( \int_0^\infty \frac{\cos a x \, dx}{(x^2+b^2)^2+c^2}\)
- \(\S\S\) 7.4 Q[10] \(\int_0^\infty \frac{x \sin ax\, dx}{(x^2+b^2)^2 + c^2}\)
- \(\S\S\) 7.4 Q[11] \(\int_0^\infty \frac{\sin^2 ax \cos^2 bx}{\beta^2+x^2} dx\)
- \(\S\S\) 7.4 Q[12] \(\int_0^\infty \frac{x\sin^2 ax \cos^2bx}{\beta^2+x^2} dx\)
- \(\S\S 7.5\) Tutorial ----- Integration Around a Branch Cut
- \(\S\S\) 7.6 Integration around a branch cut
- \(\S\S\) 7.6 Q[1] \(\int_0^\infty \frac{x^{\frac{1}{2}}}{(x+4)(x+25)}\)
- \(\S\S\) 7.6 Q[2] \(\int_0^\infty \frac{x^{p-1}}{x^2+2x+2}\)
- \(\S\S\) 7.6 Q[3] \(\int_0^\infty \frac{x^{p-1}}{(x+1)^2}\, dx\)
- \(\S\S\) 7.6 Q[4] \(\int_0^\infty \frac{x^p}{x^2+a^2}\, dx\)
- \(\S\S\) 7.6 Q[5] \(\int_0^\infty \frac{x^p}{(x^2+a^2)^2}\, dx\)
- \(\S\S\) 7.6 Q[7] \( \int_0^\infty \frac{x^p}{x^6+1}\, dx \)
- \(\S\S\) 7.6 Q[8] \( \int_0^\infty \frac{x^{1/2}}{(x^2+1)(x^2+4)}\, dx\)
- \(\S\S\) 7.6 Q[9] \(\int_0^\infty \frac{x^{p-1}}{1+x^q}\)
- \(\S\S\) 7.6 Q[11] \(\int_1^\infty \frac{(x-1)^{p-1}}{x^2}\,dx\)
- \(\S\S\) 7.6 Q[12] \(\int_1^\infty\frac{(x-1)^{p-1}}{x^3}\, dx\)
- \(\S\S\) 7.6 Q[13] \(\int_0^\infty \frac{ x^{p-1} dx}{x^3+b^3}\)
- \(\S\S\) 7.6 Q[14] \(\int_0^\infty \frac{x^p}{x^2+2x\cos\alpha +1} \)
- \(\S\S\) 7.7 Integral of Type \(\int \log x Q(x) \,dx\)
- \(\S\S\) 7.7 Q[1] \( \int_0^\infty \frac{\log x}{x^2+1}\,dx\)
- \(\S\S\) 7.7 Q[2] \(\int_0^1 \frac{\log x}{x^3+1}\)
- \(\S\S\) 7.7 Q[3] \(\int_0^\infty \frac{\log^2x}{x^2+1}\, dx\)
- \(\S\S\) 7.7 Q[4]-[6] \(\int_0^\infty \frac{\log^n x}{x^2+1}\, dx, n=4,6,8 \)
- \(\S\S\) 7.7 Q[7] \(\int_0^\infty \frac{\log(px) }{x^2+q^2}\, dx\)
- \(\S\S\) 7.7 Q[8] \(\int_0^\infty \frac{x^2\log x}{(x^2+1)^2} \, dx\)
- \(\S\S\) 7.7 Q[9] \(\int\frac{\log x}{(x^2+1)(1+q^2x^2)}\, dx\)
- \(\S\S\) 7.7 Q[11] \(\int_0^\infty \frac{\log x\, dx}{x^2+2ax\cos\alpha + a^2}\)
- \(\S\S\) 7.7 Q[12] \(\int_0^\infty \frac{x^2 \log x\, dx }{(a^2+b^2x^2)(1+x^2)}\)
- \(\S\S\) 7.8 Hyperbolic Functons
- \(\S\S\) 7.9 Exercise ----- Exponential and Hyperbolic Functions
- \(\S\S\) 7.9 Q[1] \(\int_{-\infty}^\infty \frac{x\exp(-x)}{1+\exp(-4x)}\, dx\)
- \(\S\S\) 7.9 Q[3] \(\int_{-\infty}^\infty \frac{x \exp(p x)}{1+ \exp(x)}\, dx\)
- \(S\S\) 7.9 Q[4] \(\int_{-\infty}^\infty x^2 \frac{\exp(px)}{1+ \exp(x)}\, dx\)
- \(\S\S\) 7.9 Q[5] \( \int _0^\infty x \frac{[1-\exp(-x) ]\exp(-x)}{1+ \exp(-3x)} \,dx\)
- \(\S\S\) 7.9 Q[6] \(\int_0^\infty \frac{x\{\exp(-qx) + \exp[(q-p)x]\}}{1- \exp(-px)}\, dx \)
- \(\S\S\) 7.9 Q[7]-[10] \(\int_0^\infty \frac{x^n}{\cosh \pi x} \, dx, n=2,4,6,8\)
- \(\S\S\) 7.9 Q[11] \(\int_0^\infty \frac{x \sinh ax}{\cosh bx}\, dx\)
- \(\S\S\) 7.9 Q[12] \( \int_0^\infty \frac{\cosh ax}{\cosh bx}\, dx \)
- \(\S\S\) 7.9 Q[13] \(\int_0^\infty \frac{x^2 \cosh ax}{\cosh bx}\, dx \)
- \(\S\S\) 7.9 Q[14] \(\int_0^\infty \frac{dx}{\cosh ax + \cos t} \)
- \(\S\S\) 7.10 Tutorial Principal Value Integrals
- \(\S\S\) 7.11 Exercise Integrals Requiring Indented Contours
- $ \S\S 7.11$ Q[1] $ \int_0^\infty \frac{(1-\cos a x)}{x^2}dx, \quad a >0 $
- \(\S\S 7.11\) Q[2] \(\int_0^\infty \frac{(\sin x - x\cos x)}{x^3} dx \)
- \(\S\S\) 7.11 Q[3] \( \int_0^\infty \frac{\sin^3x}{x^3} dx \)
- \(\S\S\) 7.11 Q[4] \( \int_0^\infty \frac{x^3-\sin^3x}{x^5} dx\)
- \(\S\S\) 7.11 Q[5] \( \int_0^\infty \frac{\sin (p x) \sin (q x)}{x^2} dx\)
- \(\S\S\) 7.11 Q[6] \( \int_0^\infty \frac{\sin^2 px \cos 2qx}{x^2} dx \)
- \(\S\S\) 7.11 Q[7] \(\int_0^\infty \frac{\cos px - \cos qx}{x^2} dx\)
- \(\S\S\) 7.11 Q[8] \( \int_0^\infty \frac{\sin \pi x}{x(1-x^2)} dx\)
- \(\S\S\) 7.11 Q[9] \( \int_0^\infty \frac{1}{1-x^4} dx \)
- \(\S\S\) 7.11 Q[10] \(\int_0^\infty \frac{x^{p-1}}{1-x^3} dx\)
- \(\S\S 7.12\) Series Summation and Expansion
- \(\S\S\) 7.13 What you see is not what you get
- \(\S\S\) 7.14 Integrals from Statistical Mechanics
- \(\S\S\) 7.14 Q[1] \(\int_0^\infty \frac{e^x}{(e^x+1)^2} =\frac{1}{2}\)
- \(\S\S\) 7.14 Q[2] \(\int_0^\infty \frac{e^x}{(e^x+1)}=\frac{\pi^2}{6}\)
- \(\S\S\) 7.14 Q[3] \(I_m=\int_{-\infty}^\infty \frac{x^m e^x}{(e^x+1)^2}\)
- \(\S\S\) 7.14 Q[4] \(J(k) = \int_{-\infty}^\infty \frac{e^{ikx} e^x}{(e^x+1)^2}\)
- \(\S\S\) 7.14 Q[5] \(\int_0^\infty \frac{x^3}{e^x-1}=\frac{\pi^4}{15}\)
- \(S\S\) 7.15 Alternate Routes Improper Integrals
- \(\S\S\) 7.16 Killing Two Birds with One Stone
- \(\S\S\) 7.17 Open Ended : Food for Your Thought
- \(\S\S\) 7.18 Mixed Bag : Improper Integrals
- Errata for Complex Variables --- Principles and Problem Sessions
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