Mathematics – IV Detailed Syllabus for B.Tech second year first sem is covered here. This gives the details about credits, number of hours and other details along with reference books for the course.

The detailed syllabus for Mathematics – IV B.Tech 2016-2017 (R16) second year first sem is as follows.

B.Tech. II Year I Sem. L/T/P/C

Course Code: *MC400ES 4/1/0/0

Prerequisites: Foundation course (No Prerequisites).

**Course Objectives: To learn**

- differentiation and integration of complex valued functions
- evaluation of integrals using Cauchy’s integral formula
- Laurent’s series expansion of complex functions
- evaluation of integrals using Residue theorem
- express a periodic function by Fourier series and a non-periodic function by Fourier transform
- to analyze the displacements of one dimensional wave and distribution of one dimensional heat equation

**Course Outcomes**: After learning the contents of this paper the student must be able to

- analyze the complex functions with reference to their analyticity, integration using Cauchy’s integral theorem
- find the Taylor’s and Laurent’s series expansion of complex functions
- the bilinear transformation
- express any periodic function in term of sines and cosines
- express a non-periodic function as integral representation
- analyze one dimensional wave and heat equation

**UNIT–I Functions of a complex variable:** Introduction, Continuity, Differentiability, Analyticity, properties, Cauchy, Riemann equations in Cartesian and polar coordinates. Harmonic and conjugate harmonic functions-Milne-Thompson method

**UNIT-II Complex integration:** Line integral, Cauchy’s integral theorem, Cauchy’s integral formula, and Generalized Cauchy’s integral formula, Power series: Taylor’s series- Laurent series, Singular points, isolated singular points, pole of order m – essential singularity, Residue,Cauchy Residue theorem (Without proof).

**UNIT–III Evaluation of Integrals:** Types of real integrals: (a) Improper real integrals Bilinear transformation- fixed point- cross ratio- properties- invariance of circles.

**TEXT BOOKS:**

- A first course in complex analysis with applications by Dennis G. Zill and Patrick Shanahan, Johns and Bartlett Publishers.
- Higher Engineering Mathematics by Dr. B. S. Grewal, Khanna Publishers.
- Advanced engineering Mathematics with MATLAB by Dean G. Duffy

**REFERENCES:**

- Fundamentals of Complex Analysis by Saff, E. B. and A. D. Snider, Pearson.
- Advanced Engineering Mathematics by Louis C. Barrett, McGraw Hill.

For all other B.Tech 2nd Year 1st Sem syllabus go to JNTUH B.Tech Metallurgical and Materials Engineering 2nd Year 1st Sem Course Structure for (R16) Batch.

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