Paper 6

Core VI : Mathematical Physics-III

2019 CBCS Core VIII

Syllabus

(Understanding and application of Complex function variables, properties and application of Fourier Integral transform, Laplace integral transformation.)

Unit I
  • Complex Analysis: Brief Revision of Complex Numbers and their Graphical Representation
  • Euler’s formula, De Moivres theorem, Roots of complex Numbers, Functions of Complex Variables, Analyticity and Cauchy-Riemann Conditions, Examples of analytic functions, Singular functions: poles and branch points, order of singularity, branch cuts, Integration of a function of a complex variable, Cauchys Inequality, Cauchys Integral formula, Simply and multiply connected region, Laurent and Taylors expansion, Residues and Residue Theorem,
  • Application in solving Definite Integrals.
Unit II

Integral Transforms-I: Fourier Transforms: Fourier Integral theorem, Fourier Transform, Examples, Fourier Transform of trigonometric, Gaussian, finite wave train and other functions, Representation of Dirac delta function as a Fourier Integral, Fourier transform of derivatives, Inverse Fourier Transform.

Unit III

Integral Transforms-II: Convolution theorem, Properties of Fourier Transforms
(translation, change of scale, complex conjugation), Three dimensional Fourier transforms with examples, Application of Fourier Transforms to differential equations: One dimensional Wave and Diffusion/Heat flow Equations.

Unit IV
  • Laplace Transforms: Laplace Transforms (LT) of Elementary functions,
  • Properties of Laplace Transforms: Change of Scale Theorem, Shifting Theorem, LTs of
    Derivatives and Integrals of Functions, Derivatives and Integrals of Functions, Derivatives and Integrals of LTs. LT of Unit Step function, Dirac Delta function, Periodic Functions, Inverse LT,
  • Application of Laplace Transforms to Differential Equations: Damped Harmonic Oscillator, Simple Electrical Circuits.
Text Books:
  • Mathematical Methods for Physicists, G. B. Arfken, H. J. Weber, F. E. Harris (2013, 7th Edn.,Elsevier)
  • Advanced Engineering Mathematics, ErwinKreyszig (WileyIndia)
Reference Books:
  • Mathematical Physics and Special Relativity – M. Das, P. K. Jena and B. K. Dash (Srikrishna Prakashan)
  • Mathematical Physics – H. K. Dass, Dr. Rama Verma (S. Chand Publishing)
  • Mathematical Physics - C. Harper (Prentice Hall India)
  • Complex Variable: Schaum’s Outlines Series M. Spiegel (2nd Edition , Mc-Graw Hill
    Education)
  • Complex variables and applications J.W.Brown and R.V.Churchill
  • Mathematical Physics, Satya Prakash (Sultan Chand)
  • Mathematical Physics B.D.Gupta (4thedition,VikasPublication)
  • Mathematical Physics B.S.Rajput, Pragati Prakashan
  • Mathematical Physics-III, (University Physics), Dr. Ranjan Kumar Bhuyan, Himalaya Publishing House

Questions

2025 | +3-IV-S-CBCS(MS)-Sc(H)-Core-VIII-Phy-R&B | Full Mark 60

PART-I | Answer all the questions | 1x8
  • Write down the value of $\frac{1+i}{1-i}$.
  • If $f(z)=z^$, the value of $f^(z)-f(z)$ in plane polar coordinates is _____.
  • A function is analytic at each point in the finite complex plane is _____ function.
  • The function $f(z) = \sin \frac{1}{z-z_0}$ has ____ singularity at $z=z_0$.
  • If $f(x)=f^(x)$, $F^(k)=$ ______.
  • Write down Fourier Transform in 3-Dimensions.
  • Laplace Transform of 1 is _____.
  • The inverse Laplace transform of $1/(s-k)$ is _____.
PART-II | Answer any eight within two to three sentences | 1.5 ×8
  • Find square root of 1.
  • Write down Cauchy -Riemann conditions.
  • State De Moivre's theorem.
  • Write down Cauchy integral formula.
  • Check the analyticity of $f(z)=e^z$.
  • Find the inverse Laplace transform of 1/s.
  • Prove the linearity property of fourier Transform.
  • Find the Laplace Transform of unit step function.
  • Find the Fourier Transform of Dirac $\delta$-function.
  • State the convolution theorem for Fourier Transform.
PART-III | Answer any eight of the following in maximum 75 words. | 2× 8
  • Express $-5+5i$ in polar form.
  • Find the Poles and residues of the function $f(z)= \frac{1}{z^2+1}$.
  • Find the sine transform of $e^{-ax}$.
  • Prove the change of scale property of Fourier Transform.
  • Find the Laplace Transform of $\sin {at}$.
  • Show the complex conjugation property of Fourier Transform.
  • Find Laplace Transform of $\delta (x-a)$.
  • If $1, \omega, \omega^2$ are the cube roots of unity, show that $(1-\omega+\omega^2)(1-\omega-\omega^2)=4$.
  • Find the inverse Laplace transform of $\frac{1}{ (s+ z)^4}$.
  • Explain derivatives of Fourier Transform.
PART-IV | Answer within 500 words each. | 6×4

4) State and prove Laurent's expansion theorem.

OR

State Cauchy's residue theorem. Apply it to evaluate the integral $$\int_0 ^\infty \frac{dx}{1+x^2}$$

5) Deduce the Fourier Transform of $Ne^{-ax^2}$, where N and 'a' are real constants.

OR

a. State and prove shifting theorem of Fourier transform.

b. State and prove convolution theorem of Fourier Transform.

6) Deduce Perseval's relation. How has it been useful in quantum mechanics ?

OR

Derive an expression for 1-D wave equation using derivatives of Fourier Transform and hence find its solution.

7) a. State and prove change of scale property of Laplace Transform.

b. Using Laplace transform derivatives, find Laplace transform of $\cos kt$.

OR

Study the Damped harmonic oscillator problem and obtain its solution using Laplace transform derivatives.

2023 | +3-IV-S-CBCS(MS)-Sc(H)-Core-VIII-Phy-R&B | Full Marks : 60

PART-I | Answer the following questions | 1x8

a . A function is said to be singular if it is not ______ within a certain domain or range.

b. $\int_c f(z) dz =0$ for _____ Theorem.

c. Fourier transforms are used to solve partial wave equations of _____ variables.

d. Laplace transform of $\delta (t)$ is _____.

e. Laplace transform are used to find solution of _____ valuel problem.

f. Fourier sine Transform of Dirac Delta function is _____.

g. Laplace transform of $\sin (ax)$ is given as _____.

h. Fourier Transform of a _____ is the pointwise product of fourier Transform.

PART-II | Answer any eight within two to three sentences | 1.5x8

a. Find the argument of $-3i$.

b. Find the singularity of $f(z) = 1/z$.

c. Evaluate $\oint_c \tan z dz$ where C:|z|= 1.

d. Find the fourier Transform of $\delta (x-a)$.

e. State the Linearty property of Laplace Transform.

f. Find the modulus of $(1+ \sqrt{3}i)$.

g. Find the value of $L(t^n)$.

h. What is Dirac Delta Function ?

i. What is unit step function ?

j. Write Heat flow Equation.

PART-III | Answer any eight of the following (in maximum 75 words.) | 2x8

a. State De Moivre's Theorem.

b. Explain order of singularity.

c. Explain Finite Wave Train functioh.

d. Explain Inverse Fourier Transform.

e. Explain complex conjugation.

f. State shifting Theorem of Laplace Transforms.

g. Find the Laplace Transform of $F(t) = 4t^2-3$.

h. Find the Fourier cosine transform of $f(x) = e^{-ax}$.

i. Find the Fourier sine transform of 1/x.

j. Explain the change of scale property.

PART-IV | Answer within 500 words each. | 6x4

  1. State and prove Cauchy's Residue Theorem.

    OR

    State and derive Taylor's series expansion of f(z).

  2. a. Expand $f(x) =x^2$ for $-\pi \lt x \lt \pi$ in a Fourier series.

b. Develop Fourier Transform using Fourier Integral.

OR

Find the Fourier Transform of the Gaussian distribution function $f(x)= Ae^{-ax^2}$, where 'A' and 'a' are constants.

  1. Apply Fourier Transform to solve the one - dimensional heat flow equation

    OR

    State and prove convolution Theorem.

  2. Using Laplace Transform, find the solution of differential equation of Damped Harmonic oscillator.

    OR

    Find the Inverse Laplace Transform of

a. $s^{-3/2}$.

b. $\frac{a}{s(s+a)}$.

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