MALLA REDDY COLLEGE OF ENGINEERING & TECHNOLOGY (An Autonomous Institution – UGC, Govt.of India) Recognizes under 2(f) and 12(B) of UGC ACT 1956 (Affiliated to JNTUH, Hyderabad, Approved by AICTE –Accredited by NBA & NAAC-“A” Grade-ISO 9001:2015 Certified) MATHEMATICS-III B.Tech – II Year – I Semester DEPARTMENT OF HUMANITIES AND SCIENCES
CONTENTS 1. Unit –I (Fourier Series) 1-23 2. Unit –II (Fourier Transforms) 24-41 3. Unit – III (Analytic Functions) 42-71 4. Unit-IV (Singularities and Residues) 72-98 5. Unit-V (Conformal Mappings) 99-118 24
(R18A0023)MATHEMATICS-III Objectives: To learn 1. The expansion of a given function by Fourier series. 2. The Fourier sine and cosine transforms, properties, inverse transforms, and finite Fourier transforms. 3. Differentiation, integration of complex valued functions and evaluation of integrals using Cauchy’s integral formula. 4. Taylor’s series, Laurent’s series expansions of complex functions and evaluation of integrals using residue theorem. 5. Transform a given function from z - plane to w – plane. Identify the transformations like translation, magnification, rotation, reflection, inversion, and Properties of bilinear transformations. UNIT – I: Fourier series Definition of periodic function, Fourier expansion of periodic functions in a given interval of length 2𝜋, Fourier series of even and odd functions, Half-range Fourier sine and cosine expansions, Fourier series in an arbitrary interval. UNIT – II: Fourier Transforms Fourier integral theorem - Fourier sine and cosine integrals. Fourier transforms – Fourier sine and cosine transforms, properties. Inverse transforms and Finite Fourier transforms. UNIT – III: Analytic functions Complex functions and its representation on Argand plane, Concepts of limit, continuity, differentiability, Analyticity, and Cauchy-Riemann conditions, Harmonic functions – Milne – Thompson method. Line integral – Evaluation along a path and by indefinite integration – Cauchy’s integral theorem (singly and multiply connected regions) – Cauchy’s integral formula – Generalized integral formula. UNIT – IV: Singularities and Residues Radius of convergence, expansion of given function in Taylor’s series and Laurent series. Singular point –Isolated singular point, pole of order m and essential singularity. Residues – Evaluation of residue by formula and by Laurent series. Residue theorem- Evaluation of improper integrals of the type (a) (b)
UNIT – V: Conformal Mappings Conformal mapping: Transformation of z-plane to w-plane by a function, conformal transformation. Standard transformations- Translation; Magnification and rotation; inversion and reflection, Transformations like ez , log z, z2, and Bilinear transformation. Properties of Bilinear transformation, determination of bilinear transformation when mappings of 3 points are given (cross ratio). TEXT BOOKS: i) Higher Engineering Mathematics by B.S. Grewal, Khanna Publishers. ii) Higher Engineering Mathematics by B.V Ramana , Tata McGraw Hill. iii)Advanced Engineering Mathematics by Kreyszig, John Wiley & Sons. REFERENCES: i) Complex Variables and Applications by James W Brown and Ruel Vance Churchill-Mc Graw Hill ii)Mathematics-III by T K V Iyenger ,Dr B Krishna Gandhi, S Ranganatham and Dr MVSSN Prasad, S chand Publications. iii) Advanced Engineering Mathematics by Michael Greenberg –Pearson publishers. Course Outcomes: After going through this course the students will be able to 1. Find the expansion of a given function by Fourier series in the given interval. 2. Find Fourier sine, cosine transforms and inverse transformations. 3. Analyze the complex functions with reference to their analyticity and integration using Cauchy’s integral theorem. 4. Find the Taylor’s and Laurent series expansion of complex functions. Solution of improper integrals can be obtained by Cauchy’s-Residue theorem. 5. Understand the conformal transformations of complex functions can be dealt with ease.