This book “Mathematical Physics -III” for B.Sc. Hons. students is written according to new NEP 2020 SYLLABUS of all Odisha Universities, Autonomous Colleges and other Indian Universities as per UGC guidelines. In addition, this book will serve as a text book to the students preparing for IIT JAM, NEST, JEST, CUET and CPET to take admission in M.Sc. course in physics.
The content of this book have been divided into four units and ten chapters. The subject matter of each chapter is developed systematically in a simple, lucid and logical manner for easy grasp of the students. The structure of each chapter includes the text followed by sufficient numbers of selected informative and modern solved problems to illustrate the text and exercise. The exercises are again divided into three parts: (i) multiple choice questions for quick review, (ii) large number of unsolved short and long questions for practice and (iii) many simple and moderately difficult problems for the understanding of mathematical concepts used within the text. Answers to all the multiple questions and problems for practice are provided.
While writing this book emphasis is given to present the subject matter in a simpler way without any jumping of mathematical steps and with necessary diagrams, so that an average student may feel no difficulty in understanding the text.
Contents –
UNIT – I
1. COMPLEX VARIABLES
1.1 REAL NUMBER SYSTEM – 1.2 THE COMPLEX NUMBER SYSTEM – 1.3 POWERS OF i – 1.4 BASIC ARITHEMATIC OPERATIONS WITH COMPLEX NUMBERS – 1.5 BASIC ALGEBRAIC PROPORTIES– 1.6 COMPLEX CONJUGATE – 1.7 MODULUS OF A COMPLEX NUMBER – 1.8 THE COMPLEX PLANE – 1.8.1 Graphical Representation Of Complex Numbers – 1.8.2 Polar Representation Of Complex Numbers – 1.8.3 Working rule to find the principal argument [Arg (z)] of a complex number – 1.9 EXPONENTIAL FORM OF Z – 1.10 EULER’S FORMULA – 1.11 DE-MOIVRE’S FORMULA (By Exponential Function) – 1.12 DE-MOIVRE’S THEOREM (By induction) – 1.13 VECTOR REPRESENTA-TION OF COMPLEX NUMBERS AND THEIR COMPOSITION – 1.14 nth ROOTS OF A COMPLEX NUMBER – 1.15 FUNCTIONS OF COMPLEX VARIABLE – 1.16 COMPLEX FUNCTIONS AS MAPPING – 1.17 SINGLE-VALUED AND MULTIVALUED FUNCTIONS – 1.18 MULTIVALUED FUNCTIONS, BRANCH POINT AND BRANCH CUT – 1.19 SOME ELEMENTARY FUNCTIONS OF COMPLEX VARIABLES.
Multiple Choice Questions
Questions
Problems for Practice
2. LIMITS AND CONTINUITY
2.1 LIMIT OF A FUNCTION OF A COMPLEX VARIABLE – 2.1.1 Non existence of a complex limit – 2.2 THEOREMS ON LIMITS – 2.3 LIMITS INVOLVING THE POINT OF INFINITY – 2.4 METHODS OF EVALUATION OF LIMITS – 2.5 CONTINUITY – 2.5.1 Properties of continuous functions – 2.5.2 Continuity of a function at z → ∞ – 2.5.3 Theorems on continuity – 2.5.4 Discontinuity of a function.
Multiple Choice Questions
Questions
Problems for Practice
3. COMPLEX DIFFERENTIATION AND ANALYTIC FUNCTION
3.1 DERIVATIVE OF A COMPLEX FUNCTION – 3.1.1 Derivative Rules – 3.1.2 Continuity and Differentiability – 3.2 ANALYTIC FUNCTIONS – 3.3 CAUCHY-RIEMANN EQUATIONS – 3.4 CAUCHY-RIEMANN EQUATION IN POLAR FORM – 3.5 HARMONIC FUNCTIONS – 3.6 HARMONIC CONJUGATES – 3.7 METHOD TO FIND THE HARMONIC CONJUGATES.
Multiple Choice Questions
Questions
Problems for Practice
4. COMPLEX INTEGRATION (Cauchy’s Integral Theorem and Formula)
4.1 COMPLEX LINE INTEGRALS (RIEMAN’S DEFINITION OF INTEGRATION)
– 4.1.1 Reduction of ∫ f(z) dz to real integrals – 4.1.2 ∫ f(z) dz expressed as a definite integral – 4.2 FUNDAMENTAL THEOREM FOR
COMPLEX LINE INTEGRALS – 4.3 PROPERTIES OF COMPLEX LINE INTEGRATS – 4.4 ESTIMATION OF ABSOLUTE VALUE OF A COMPLEX LINE INTEGRALS – 4.5 SOME IMPORTANT DEFINITIONS – 4.6 CAUCHY INTEGRAL THEOREM – 4.6.1 Extension of Cauchy’s theorem to multiple connected region – 4.6.2 Consequences of Cauchy’s Theorem – 4.7 CAUCHY’S INTEGRAL FORMULA – 4.8 CONSEQUENCES OF CAUCHY’S INTEGRAL FORMULA – 4.8.1 Derivatives of Analytic Functions – 4.8.2 Existence of Derivatives of Analytic Function – 4.8.3 Cauchy’s Inequality – 4.9 LIOUVILLE THEOREM.
Multiple Choice Questions
Questions
Problems for Practice
5. TAYLOR AND LAURENT SERIES (Singularities)
5.1 GEOMETRIC SERIES – 5.2 CONNECTION TO CAUCHY’S INTEGRAL FORMULA – 5.3 POWER SERIES – 5.3.1 Convergence of Power Series – 5.3.2 Integration and Differentation of Power Series – 5.3.3 Radius of Convergence of Power Series – 5.4 TAYLOR’S THEOREM – 5.4.1 Taylor Series of Some Elementary Functions around z0 = 0 – 5.5 THE LAURENT SERIES – 5.6 MODIFIED FORM OF LAURENT SERIES – 5.7 ZEROS OF AN ANALYTIC FUNCTION – 5.8 SINGULARITIES – 5.8.1 Classification of Singularities.
Multiple Choice Questions
Questions
Problems for Practice
6. RESIDUES, RESIDUE THEOREM AND ITS APPLICATIONS
6.1 RESIDUE – 6.2 METHODS OF FINDING RESIDUES – 6.2.1 Residue at simple pole – 6.2.2 Residue at a multiple pole – 6.2.3 Residue at a pole at z = z0 of any order – 6.2.4 Residue at infinity – 6.3 CAUCHY’S RESIDUE THEOREM – 6.4 APPLICATIONS OF RESIDUE THEOREM – 6.4.1 Evaluation of definite real integrals.
Multiple Choice Questions
Questions
Problems for Practice
UNIT – II
7. INTEGRAL TRANSFORMS-I
7.1 FOURIER INTEGRAL THEOREM – 7.2 FOURIER COSINE AND SINE INTEGRALS – 7.3 COMPLEX FORM OF FOURIER INTEGRAL – 7.4 FOURIER TRANSFORM AND ITS INVERSE – 7.5 FOURIER AND SINE TRANSFORM – 7.6 FOURIER TRANSFORM OF DERIVATIVES – 7.7 FOURIER SINE AND COSINE TRANSFORM OF DERIVATIVES – 7.7.1 Fourier sine transform of the 2nd derivative of a function f(x) – 7.7.2 Fourier cosine transform of the 2nd derivative of a function f(x) – 7.8 FINITE FOURIER TRANSFORM.
Multiple Choice Questions
Questions
Problems for Practice
UNIT – III
8. INTEGRAL TRANSFORMS-II
8.1 PROPERTIES OF FOURIER TRANSFORM – 8.2 CONVOLUTION – 8.3 PARSEVAL’S RELATION – 8.4 FOURIER TRANSFORM OF DERIVATIVES – 8.5 FOURIER TRANSFORM PAIR IN 2D – 8.6 FOURIER TRANSFORM PAIR IN 3D – 8.7 APPLICATION OF FOURIER TRANSFORM TO DIFFERENTIAL EQUATIONS – 8.8 FOURIER SINE/COSINE TRANSFORM APPLIED TO THE HEAT EQUATION.
Multiple Choice Questions
Questions
Problems for Practice
UNIT – IV
9. LAPLACE TRANSFORMS
9.1 DEFINITION OF LAPLACE TRANSFORM – 9.2 LAPLACE TRANSFORMS OF SOME ELEMENTARY FUNCTIONS – 9.3 EXISTENCE THEOREM FOR LAPLACE TRANSFORM – 9.4 PROPERTIES OF THE LAPLACE TRANSFORM – 9.5 LAPLACE TRANSFORMS OF DERIVATIVES AND INTEGRALS – 9.5.1 Laplace Transform of Derivatives – 9.5.2 Laplace Transform of Integrals – 9.6 LAPLACE TRANSFORM OF SOME SPECIAL FUNCTIONS – 9.6.1 The Unit Step Function – 9.6.2 Laplace Transform of Uniform Step Function – 9.7 DIRAC DELTA FUNCTION – 9.7.1 Laplace Transform of Dirac-Delta Function – 9.8 PERIODIC FUNCTIONS – 9.8.1 Laplace Transform of Periodic Functions.
Multiple Choice Questions
Questions
Problems for Practice
10. INVERSE LAPLACE TRANSFORMS (Solution of Differential Equation)
10.1 INVERSE LAPLACE TRANSFORM – 10.2 INVERSE LAPLACE TRANSFORM OF SOME ELEMENTARY FUNCTIONS – 10.3 PROPORTIES OF INVERSE LAPLACE TRANSFORM – 10.4 INVERSE LAPLACE TRANSFORM USING PARTIAL FRACTION METHOD – 10.5 INVERSE LAPLACE TRANSFORM BY PARTIAL FRACTION METHOD – 10.6 CONVOLUTION THEOREM – 10.7 HEAVISIDE INVERSE FORMULA – 10.8 SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS – 10.9 USE OF LAPLACE TRANSFORM – 10.10 APPLICATION OF LAPLACE TRANSFROM TO ELECTRICAL CIRCUIT.
Multiple Choice Questions
Questions
Problems for Practice
