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Comments embedded in array of equationsNode id: 253page |
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Commutator Node id: 283page |
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Complete Orthonormal SetsNode id: 812page |
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Complex Variables Node id: 5619forum |
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Complex Variables Node id: 2989page |
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Complex Variables --- Principles and Problem Sessions :: Top-PageNode id: 2436pageThis section of the contents will have solutions to Problems in Part-II of the book
A. K. Kapoor, "Complex Variables --Principles and Problem Sessions"
World Scientific Publshers, Singpore (2011);
Low priced edition by Cambridge University Press
Errata to the above book will also appear in this tree hierarchy.
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Complex Variables --- Principles and Problem Sessions === SOLUTIONS and ERRATANode id: 2193curated_contentIt is proposed to post solutions to almost all the problem session in the book here. At present solutions to all most all Problem Sessions of Chapter 7, Contour Integration, has been completed. The solutions to problems in other chapters is being taken up.
At present solutions to problems in Chapter 3. "Function with Branch Point Singularity"
is Being Uploaded.
Errata is being compiled and will be uploaded in this book hierarchy.
Click to see what is available.
I take this opportunity to thank my friend and colleague, Prof. T. Amarnath, from School of Mathematics, University of Hyderabad for his kind words of appreciation and for recommending the book to National Board of Higher Mathematics for inclusion in scheme of distribution of mathematics books to Universities and Institutes in India.
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Complex Variables --- Mostly contour integrationNode id: 3032pageThe problem sessions included here are in addition to those in my book on complex variables.
At present most problem sets are on method of contour integration for improper integrals.
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Complex Variables -- Problem SetsNode id: 2949page |
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Complex Variables :: Contour Integration Node id: 2960page Here we list a sample of kinds of integrals that can be evaluated by the method of contour integration. Each problem requires a different method.
A link is provided where you can find more examples.
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Complex-Variables-Home [CV-HOME] [LNK]Node id: 3488collectionComplex Variables --- Principles and Problem Sessions SOLUTIONS and ERRATA
- \(\S\S\) 4.1 Questions Range of Parameters In Improper Integrals
- \(\S\S\) 4.2 Tutorial Computing Line Integrals in Complex Plane
- \(\S\S\) 4.3 Exercise Evaluation of Line Integrals
- \(\S\S\) 4.4 Questions Deformation of Contours
- \(\S\S\) 4.5 Exercise Deformation of Contours
- \(\S\S\) 4.6 Exercise Cauchy Theorem
- \(\S\S\) 4.7 Tutorial Shift of Real Integration Parameter by a Complex Number
- Q[1] Prove that \(\int_0^\infty exp(-px^2) \cos(2bx)\, dx= \frac{1}{2}\sqrt{\frac{\pi}{p}} \exp(-b^2/p), p>0 \)
- Q[2] Prove that \(\int_0^\infty e^{-px^2} \sin(ax) \sin(bx) = \frac{1}{4} \left\{\exp\Big(\frac{(a-b)^2}{4p}\Big)- \exp\Big( -\frac{-(a+b)^2}{4p}\Big) \right\}\)
- \(\S\S\) 4.8 Tutorial Scaling of a Real Integration Variable by a Complex Number
- \(\S\S\) 4.9 Exercise Shift and Scaling by a Complex Number
- Q[1] \(\int_0^\infty e^{-q^2x^2} \sin^2 ax\, dx\)
- Q[2] \(\int_0^\infty e^{-q^2x^2}{\,\sin(px+c)\,\choose\, \cos (px+c)\,}\, dx \)
- Q[3] \( \int_0^\infty x e^{-q^2x^2} \sin ax \, dx\)
- Q[4] \( \int_0^\infty x^2 e^{-q^2x^2} \cos ax\, dx\)
- Q[5] \(\int_0^\infty x^3 e^(-q^2x^2) \sin ax \, dx \)
- Q[6] \(\int xe^{-px} \cos (2x^2+px) \, dx\)
- Q[7] \(\int xe^{-px} \cos (2x^2-px) \, dx\)
- Q[8] \(\int_0^\infty e^{-px} \big[ \sin(2x^2+px)+\cos(\cos (2x^2 +px) ) \big]\, dx\)
- Q[9] \( \int_0^\infty e^{-px} \big[ \sin(2x^2-px)-\cos(\cos (2x^2 -px) ) \big]\, dx\)
- Q[10] \(\int_0^\infty xe^{-\beta x} \sin ax^2 \sin\beta x\, dx\)
- Q[11] \(\int_0^\infty \cos ax^2 \cos 2bx, dx\)
- Q[12] \(\int_0^\infty (\cos ax +\sin ax) \sin(bx)^2\, dx\)
- Q[13] \(\int_0^\infty \sin ax^2 \sin 2bx \sin 2cx\, dx\)
- Q[14] \(\int_0^\infty x \sin ax^2 \sin2bx \,dx\)
- Q[15] \(\int_0^\infty \sin ax^2 \cos bx^2\, dx\)
- Q[16] \(\int_0^\infty (\sin^2 ax^2 -\sin^2bx^2)\, dx\)
- Q[17] \(\int_0^\infty \big(\sin (a-x^2) + \cos(a-x^2) \big)\, dx\)
- \(\S\S\) 4.10 Exercise Rotation of Contour
- \(\S\S\) 4.11 Mixed Bag
- \(\S\S\) 5.2
- \(\S\S\) 5.3
- \(\S\S\) 5.5
- \(\S\S\) 5.4 :: Taylor Series Representation
- \(\S\S\) 5.8 :: Laurent Expansion of \(f(z)\) in powers of \((z-z_0)
- 5.8 Q[1] \(f(z) = (z^2+3z+5) e^{1/z}, z_0=0.\)
- 5.8 Q[2] \(f(z) = \frac{\text{cosh}\, z}{z(z^4+1)}, z_0=0.]\)
- 5.8 Q[3] \(f(z) = z^2 \sin(1/(1-z)), z_0=1.\)
- 5.8 Q[4] \(f(z) = \frac{z}{\text{cosh}\, z -1}, z_0=0. \)
- 5.8 Q[5] \(f(z) = \cot z , z_0=0. \)
- 5.8 Q[6] \(f(z) = \text{cosech}\, z, z_0=0. \)
- 5.8 Q[7] \(f(z) = \text{sec } z, z_0=\pi/2.\)
- 5.8 Q[8] \(f(z)=\frac{z+3}{\sin z^2}, z_0=0.\)
- 5.8 Q[9]\(f(z)=\frac{z}{\sin^3 z}, z_0=n\pi.\)
- 5.8 Q[10]\(f(z)= \frac{z}{\sin\pi z^2}, z_0=1.\)
- 5.8 Q[11]\(f(z)= \frac{1}{(\exp(z)-1)}, z_0=0.\)
- 5.8 Q[12] \(\frac{1}{(\exp(z)+1)^2)}, z_0=i\pi\)
- 5.8 Q[13] Prove \(\exp(t(z+1/z))= \sum_{-\infty}^\infty z^n J_n(t).\)
- 5.8 Q[14]\(f(z) = \sin(\pi/z), z_0=1.\)
- \(\S\S\) 6.10 Integrals Using the Residue at Infinity
- Some general, common remarks
- 6.10 Q[1] \(\oint \sqrt{(z-a)(z-b)}\, dz= -(\pi i/4)(a-b)^2\)
- 6.10 Q[2] \(\oint \sqrt{\frac{z-a}{z-b}\, dz} = -(\pi i/2)(a-b)\)
- 6.10 Q[3] \(\oint \frac{z}{\sqrt{z-a}{z-b}}\, dz = \pi i (a+b)\)
- 6.10 Q[5] \(\int_0^1 \frac{x^{1-p}(1-x)^p}{(1+x)^3}\, dx\)
- 6.10 Q[6] \(\int_0^1\frac{x^{1-p}(1-x)^p}{1+x^2}\, dx\)
- 6.10 Q[7] \(\int_0^1 \frac{x^{1-p}(1-x)^p}{(1+x^2)^2}\, dx\)
- 6.10 Q[8] \(\int_0^1 \frac{1}{(1+x)^3(x^2(1-x))^{1/3}}\, dx\)
- 6.10 Q[9] \(\int \frac{x^{2n}}{(1+x^2)\sqrt{1-x^2}}\, dx\)
- 6.10 Q[10] \(\int_0^1 \frac{x^{2n}}{(x(1-x^2))^{1/3}}\)
- 6.10 Q[11] \(\int_0^1 \frac{1}{(1-x^n)^{1/n}}\, dx\)
- \(\S\S\) 7.1 Tutorial Integrals of Rational Function
- \(\S\S\) 7.2
- \(\S\S\) 7.3 Exercise
- \(\S\S\) 7.4 Exercise ---Integrals of \(\sin x / \cos x\) with rational functions.
- \(\S\S\) 7.5 Tutorial Integration Around a Branch Cut
- \(\S\S\) 7.6 Integration around a branch cut
- \(\S\S\) 7.7 Integral of Type \(\int
- \(\S\S\) 7.8 Hyperbolic Functions
- \(\S\S\)7.9 Exercise ----- Exponential and Hyperbolic Functions
- \(\S\S\) 7.10 Tutorial Principal Value Integrals
- \(\S\S\) 7.11 Exercise Integrals Requiring Indented Contours
- \(\S\S\) 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
- \(\S\S\) 7.15 Alternate Routes Improper Integrals
- \(\S\S\)7.16 Killing Two Birds with One Stone
- \(\S\S\) 7.17
- \(\S\S\) 7.18 Mixed Bag : Improper Integrals
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Computing Cross sectionNode id: 3544pageQuestion
Consider a system of two real scalar fields \(\phi_1, \phi_2\)
described by the Lagrangian density
\begin{equation}
\Lsc = \frac{1}{2} \partial_\mu \phi_i \partial^\mu \phi_i - \frac{1}{2} m^2
\phi_i\phi_i -\frac{1}{4} \lambda (\phi_i\phi_i)^2.
\end{equation}
Compute the scattering cross section to the lowest order in \(\lambda\). Find
the cross sections for the three processes
- \(\phi_1 + \phi_2 \longrightarrow \phi_1 +\phi_2\)
- \(\phi_1 + \phi_1 \longrightarrow \phi_1 +\phi_1\)
- \(\phi_1 + \phi_1 \longrightarrow \phi_2 +\phi_2\)
Write your answers as a constant times \(\sigma_0\equiv \frac{\lambda^2}{64\pi
s}\) where
\(s\) is the total energy in the center of mass frame.
Answer : a. \(4\sigma_0\) b. \(36 \sigma_0\) c. \(4 \sigma_0\)
Hint : Find symmetry appropriate factor.
Remark : Good question
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Concepts in ThermodynamicsNode id: 5280multi_level_page |
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Conductors and dielectrics in an electric fieldNode id: 1315page$\renewcommand{\vec}[1]{\mathbf{#1}}$
- Electric field strength near the surface of a conductor in vacuum: $$E_n = \frac{\sigma}{{\varepsilon}_0}$$
- Flux of polarization $\vec{P}$ across a closed surface: $$\oint \vec{P}.d\vec{S}=-q^\prime,$$ where $q^\prime$ is the algebraic sum of bound charges enclosed by this surface.
- Vector $\vec{D}$ and Gauss's theorem for it: $$\vec{D}=\varepsilon \vec{E}+\vec{P}, \oint \vec{D}.d\vec{S}=q,$$ where $q$ is the algebraic sum of extraneous charges inside a closed surface.
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Constant electric field in vacuumNode id: 1314page$\renewcommand{\vec}[1]{\mathbf{#1}}$
- Strength and potential of the field of a point charge $q$:$$ \vec{E}=\frac{1}{4\pi {\varepsilon}_0}\frac{q}{r^3}\vec{r}, \phi=\frac{1}{4\pi {\varepsilon}_0}\frac{q}{r}.$$
- Relation between field strength and potential:$$\vec{E}=- \vec{\nabla}\phi. $$ i.e. field strength is equal to the anti gradient of the potential.
- Gauss's theorem and circulation of the vector $\vec{E}$:$$\oint \vec{E}.d\vec{S}=\frac{q}{{\varepsilon}_0}, \oint \vec{E}.d\vec{r}=0.$$
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Constant magnetic field, magneticsNode id: 1318page$\renewcommand{\vec}[1]{\mathbf{#1}}$
- Magnetic field of a point charge $q$ moving with non-relativistic velocity $\vec{v}$: $$\vec{B}=\frac{\mu_0}{4\pi}\frac{q\, \vec{v}\times \vec{r}}{r^3}.$$
- Biot-Savart law: $$d\vec{B}=\frac{\mu_0}{4\pi} \frac{\vec{j}\times\vec{r}}{r^3}dV, d\vec{B}=\frac{\mu_0}{4\pi}\frac{I, d\vec{l}\times\vec{r}}{r^3}.$$
- Circulation of a vector $\vec{B}$ and Gauss's theorem for it: $$\oint \vec{B}.d\vec{r}=\mu_0 I, \oint \vec{B}.d\vec{S}=0.$$
- Lorentz force:$$\vec{F}=q\vec{E}+q\vec{v}\times\vec{B}.$$
- Ampere force:$$d\vec{F}=\vec{j}\times\vec{B}dV, d\vec{F}=I d\vec{I}\times\vec{B}.$$
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Content Creation Activity --- Creation of Building Blocks Node id: 6087page |
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Content Listing Index on TypeNode id: 675page |
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Contributed Problem Sets Node id: 5337collection
- 2019Classical Mechanics
- 2019CM/HMW-01
- 2019CM/HMW-02
- 2019CM/HMW-03
- 2019CM/HMW-04
- 2019CM/HMW-05
- 2019CM/HMW-07
- 2019CM/HMW-08
- 2019CM/QUIZ-01
- 2019CM/QUIZ-02
- 2019CM/QUIZ-03
- 2019CM/QUIZ-04
- 2019CM/TEST-01
- 2019CM/EXM-01
- 2018 Classical Mechanics
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Coordinate and momentum representationNode id: 1087page |
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