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Complex Variables --- Principles and Problem Sessions SOLUTIONS and ERRATA
Chapter-1 Complex Numbers
Chapter-2 :: Elementary Functions and Differentiation
Chapter 3 :: Functions with branch cut singularity
- 3..1 Questions ----------- Branch Point
- 3..2 Tutorial ---The square root branch cut
- 3.3 Exercise Branch cut for
- 3.4 Quiz Discontinuity and branch cut
- 3.5 Exercise The Logarithmic function
- 3.6 Discontinuity across the branch cut
- 3.8 Mixed Bag Multivalued Functions
- 3.8 Mixed Bag Q[1]
- 3.8 Mixed Bag Q[2]
- 3.8 Mixed Bag Q[3]
- 3.8 Mixed Bag Q[4]
- 3.8 Mixed Bag Q[5]
- 3.8 Mixed Bag Q[7]
- 3.8 Mixed Bag Q[8]
- 3.8 Mixed Bag Q[11]
- 3.8 Mixed Bag Q[12]
- 3.8 Mixed Bag Q[13]
Chapter 4 :: Integration in Complex Plane
- \(\S\S\) 4.1 Questions Range of Parameters In Improper Integrals
- Q[1] \(\int_{-\infty}^\infty \frac{}dx{(x^4+1)^c} \), Q[2], Q[3]
- Q[4] \(\\int_0^\infty \frac{x^c}{\sinh x}
- Q[5] \(\int_0^\infty \frac{e^{cx}}{\cosh^2x}\),
- Q[7] \(\int_{-\infty}^\infty \frac{x^{2n+1}{(x^2+1)}\); Q[8]; Q[9]
- Q[10] \(\int_0^\infty\frac{x^c(e^{ax}-1)}{(x^3+2)^c} \)
- Q[11] \( \int_0^\infty \frac{x^2}{e^x-1}; Q[12} ; Q{13]; Q[14] \)
- \(\S\S\) 4.2 Tutorial Computing Line Integrals in Complex Plane
- Q[1] \(\int_{x_1}^{x_2} \exp(\lambda x) dx = \frac{1}{\lambda}\[\exp(\lambda x_2) -\exp(\lambda x_2) \]\)
- Q[3] \(\int_\Gamma z^2 dz\)
- \(\S\S\) 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
Chapter 5 :: Cauchy Integral Formula
- \(\S\S\) 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]\(f(z) = \sin(\pi/z), z_0=1.\)
Chapter 6 :: The Residue Theorem
- \(\S\S\) 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] \(\int_0^1 \frac{1}{(1-x^n)^{1/n}}\, dx\)
Chapter 7 :: Contour Integration
- \(\S\S\) 7.1 Tutorial Integrals of Rational Function
- \(\S\S\) 7.2
- \(\S\S\) 7.3 Exercise
- 7.3 Q[1] \(\int_0^\infty\frac{dx}{(x^2+p^2)^2} \)
- 7.3 Q[2] \(\int_0^\infty \frac{x\, dx}{x^4+1}\)
- 7.3 Q[4]\(\int_{-\infty}^\infty \frac{dx}{(x^2+x+1)^2}\)
- 7.3 Q[5] \(\int_{-\infty}^{\infty} \frac{(x^2+2) dx}{(x^4+10x^2+9)}\)
- 7.3 Q[6] \(\int_0^\infty \frac{(x-2)dx}{(x^4+10x^2+9)}\)
- 7.3 Q[7] \(\int_0^\infty \frac{x^2\, dx}{x^6+1}\)
- 7.3 Q[8] \(\int_0^\infty \frac{dx}{(x^2+p^2)(x^2+q^2)}\)
- 7.3 Q[10] \(\int_0^\infty \frac{x^2\, dx}{x^{2N+1}+1}\)
- 7.3 Q[11] \(\int_0^\infty \frac{x^4\, dx}{x^{2N}+1}\)
- 7.3 Q[12] \(\int_0^\infty \frac{dx}{x^N+1}\)
- 7.3 Q[13] \(\int_0^\infty \frac{x^{2N}\,dx}{x^{2M}+1}\)7.3 Q[14] \(\int_0^\infty \frac{dx}{(p+qx^2)^N}\)
- \(\S\S\) 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
- 7.6 Q[1]
- 7.6 Q[2]
- 7.6 Q[3]
- 7.6 Q[4]
- 7.6 Q[5]
- 7.6 Q[7]
- 7.6 Q[8]
- 7.6 Q[9]
- 7.6 Q[11]
- 7.6 Q[12]
- 7.6 Q[13]
- 7.6 Q[14]
- \(\S\S\) 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
- 7.11 Q[1]
- 7.11 Q[2]
- 7.11 Q[3]
- 7.11 Q[4]
- 7.11 Q[5]
- 7.11 Q[6]
- 7.11 Q[7]
- 7.11 Q[8]
- 7.11 Q[9]
- 7.11 Q[10]
- \(\S\S\) 7.12 Series Summation and Expansion
- \(\S\S\) 7.13 What you see is not what you get
- 7.13 Q[1]
- 7.13 Q[2]
- 7.13 Q[3]
- 7.13 Q[4]
- 7.13 Q[5]
- 7.13 Q[6]
- 7.13 Q[7]
- 7.13 Q[8]
- 7.13 Q[9]
- 7.13 Q[10]
- 7.13 Q [11]
- 7.13 Q[12]
- \(\S\S\) 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
Errata for Complex Variables ----Principles and Problem Sessions
- Errata--Part-I :: Chapters 1-2
- Errata -- Part-I:: Chapter 4 Integration in Complex Plane
- Errata -- Part-I :: Chapter 5 Cauchy's Integral Formula
- Errata -- Part-I :: Chapter 6 Residue Theorem
- Errata -- Part-II :: \(\S\S\) 1.9Mixed Bag
- Errata -- Part-II ::\(\S\S\) 3.6Exercise Discontinuity across the branch cut
This Collection has
(a) Solutions for Problem Sessions and
(b) Errata
for my book on complex variables.
PLEASE EXPAND THE TABLE OF CONTENT AND CLICK ON SECTION OF INTEREST
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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Starting with this page the solutions to the problem sessions in the book |
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Complex Variables --- Principles and Prolems Sessions |
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