Utkal University, Odisha, Physics Semester 4, Mathematical Physics-III Syllabus

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Unit - 1 Complex Analysis

Unit 1

Complex Analysis

1.1 Brief revision of complex numbers and their graphical representation

1.2 Euler’s Formula

1.3 De Moivre’s theorem

1.4 Roots of complex numbers

1.5 Functions of complex numbers

1.6 Analyticity and CauchyRiemann Conditions

1.7 Examples of analytical function

1.8 Singular functions poles and branch points Order of singularity

1.9 Branch cuts

1.10 Integration of a function of a complex variable Cauchys inequality Cauchys integral formula

1.11 Simply and multiply connected region

1.12 Laurent and Taylors expansion

1.13 Residues and residue theorem

1.14 Application in solving simple definite integrals

Unit 1

Complex Analysis

1.1 Brief revision of complex numbers and their graphical representation

1.2 Euler’s Formula

1.3 De Moivre’s theorem

1.4 Roots of complex numbers

1.5 Functions of complex numbers

1.6 Analyticity and CauchyRiemann Conditions

1.7 Examples of analytical function

1.8 Singular functions poles and branch points Order of singularity

1.9 Branch cuts

1.10 Integration of a function of a complex variable Cauchys inequality Cauchys integral formula

1.11 Simply and multiply connected region

1.12 Laurent and Taylors expansion

1.13 Residues and residue theorem

1.14 Application in solving simple definite integrals

Unit - 2 Integral Transforms-I

Unit 2

Integral TransformsI

2.1 Fourier transform Fourier integral theorem

2.2 Fourier transform examples

2.3 Fourier transform of trigonometric Gaussian finite wave train and other functions

2.4 Representation of Dirac delta function as a fourier integral

2.5 Fourier transform of derivatives

2.6 Inverse fourier transform

Unit 2

Integral TransformsI

2.1 Fourier transform Fourier integral theorem

2.2 Fourier transform examples

2.3 Fourier transform of trigonometric Gaussian finite wave train and other functions

2.4 Representation of Dirac delta function as a fourier integral

2.5 Fourier transform of derivatives

2.6 Inverse fourier transform

Unit - 3 Integral Transforms-II

Unit 3

Integral TransformsII

3.1 Convolution theorem

3.2 Properties of fourier transform.

3.3 Three dimensional fourier transform with examples

3.4 Application of fourier transforms to differential equations 1D wave and diffusion heat flow equations

Unit 3

Integral TransformsII

3.1 Convolution theorem

3.2 Properties of fourier transform.

3.3 Three dimensional fourier transform with examples

3.4 Application of fourier transforms to differential equations 1D wave and diffusion heat flow equations

Unit - 4 Laplace Transforms

Unit – 4

Laplace Transforms

4.1 Laplace Transforms LT of Elementary functions

4.2 Properties of LT

4.3 Change of scale theorem

4.4 Shifting theorem

4.5 LTs of derivatives and integrals of functions

4.6 LT of unit step function

4.7 Dirac delta function

4.8 Periodic function

4.9 Inverse LT

4.10 Application of Laplace transform to differential equations

4.11 Damped harmonic oscillator

4.12 Simple electrical circuits

Unit – 4

Laplace Transforms

4.1 Laplace Transforms LT of Elementary functions

4.2 Properties of LT

4.3 Change of scale theorem

4.4 Shifting theorem

4.5 LTs of derivatives and integrals of functions

4.6 LT of unit step function

4.7 Dirac delta function

4.8 Periodic function

4.9 Inverse LT

4.10 Application of Laplace transform to differential equations

4.11 Damped harmonic oscillator

4.12 Simple electrical circuits

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