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Complex Variables --- Principles and Problem Sessions  SOLUTIONS and ERRATA

### Chapter 4 :: Integration in Complex Plane

• $$\S\S$$ 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
• $$\S\S$$ 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) ### Chapter 6 :: The Residue Theorem ### Chapter 7 :: Contour Integration • \(\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

### Errata for Complex Variables ----Principles and Problem Sessions

This Collection has
(a) Solutions for Problem Sessions and
(b) Errata
for my

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.

 Starting with this page the solutions to the problem sessions in the book Complex Variables --- Principles and Prolems Sessions published by World Scientific Singapore (2011) are being made available for reference
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