CSIR NET Mathematical Science UNIT – 2 Syllabus

UNIT – 2 syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for UNIT – 2 in CSIR NET Mathematical Science, not a summary of it.

1. 1

   Complex Analysis

   19 topics

   • Algebra of Complex Numbers
   • The Complex Plane
   • Polynomials
   • Power Series
   • Transcendental Functions
     • Exponential Functions
     • Trigonometric Functions
     • Hyperbolic Functions
   • Analytic Functions
   • Cauchy-Riemann Equations
   • Contour Integral
   • Cauchy’s Theorem
   • Cauchy’s Integral Formula
   • Liouville’s Theorem
   • Maximum Modulus Principle
   • Schwarz Lemma
   • Open Mapping Theorem
   • Taylor Series
   • Laurent Series
   • Calculus of Residues
   • Conformal Mappings
   • Mobius Transformations
2. 2

   Algebra

   15 topics

   • Permutations
   • Combinations
   • Pigeon-hole Principle
   • Inclusion-Exclusion Principle
   • Derangements
   • Fundamental Theorem of Arithmetic
   • Divisibility in Z
   • Congruences
   • Chinese Remainder Theorem
   • Euler’s Ø-Function
   • Primitive Roots
   • Groups
     • Subgroups
     • Normal Subgroups
     • Quotient Groups
     • Homomorphisms
     • Cyclic Groups
     • Permutation Groups
     • Cayley’s Theorem
     • Class Equations
     • Sylow Theorems
   • Rings
     • Ideals
     • Prime and Maximal Ideals
     • Quotient Rings
     • Unique Factorization Domain
     • Principal Ideal Domain
     • Euclidean Domain
   • Polynomial Rings and Irreducibility Criteria
   • Fields
     • Finite Fields
     • Field Extensions
     • Galois Theory
3. 3

   Topology

   6 topics

   • Basis
   • Dense Sets
   • Subspace and Product Topology
   • Separation Axioms
   • Connectedness
   • Compactness

UNIT – 2 flashcards for CSIR NET Mathematical Science

22 of 50 cards from the UNIT – 2 deck — real questions with worked answers.

1. State the algebraic form of a complex number and identify its real and imaginary parts.

   A complex number is written $z = x + iy$ where $x,y \in \mathbb{R}$ and $i^{2} = -1$. Here $x = \operatorname{Re}(z)$ is the real part and $y = \operatorname{Im}(z)$ is the imaginary part.

2. How are two complex numbers $z_1 = a+bi$ and $z_2 = c+di$ multiplied?

   $$z_1 z_2 = (a+bi)(c+di) = (ac - bd) + (ad + bc)i.$$

3. Define the complex conjugate of $z = x + iy$ and give two key identities involving it.

   The conjugate is $\bar{z} = x - iy$. Then $z\bar{z} = x^{2}+y^{2} = |z|^{2}$, and $\operatorname{Re}(z) = \frac{z+\bar{z}}{2}$, $\operatorname{Im}(z) = \frac{z-\bar{z}}{2i}$.

4. Give the formula for the modulus of $z = x+iy$ and state the triangle inequality.

   Modulus: $|z| = \sqrt{x^{2}+y^{2}}$. Triangle inequality: $|z_1 + z_2| \leq |z_1| + |z_2|$, with $\big||z_1| - |z_2|\big| \leq |z_1 - z_2|$.

5. Write the polar (modulus–argument) form of a nonzero complex number.

   $$z = r(\cos\theta + i\sin\theta) = r e^{i\theta},$$ where $r = |z|$ and $\theta = \arg z$ is the argument.

6. State De Moivre's theorem.

   For integer $n$, $$(\cos\theta + i\sin\theta)^{n} = \cos(n\theta) + i\sin(n\theta).$$ Equivalently $(re^{i\theta})^{n} = r^{n}e^{in\theta}$.

7. Give the formula for the $n$ distinct $n$-th roots of a complex number $z = re^{i\theta}$.

   $$z^{1/n} = r^{1/n}\exp\!\left(i\,\frac{\theta + 2\pi k}{n}\right), \quad k = 0,1,\dots,n-1.$$ They lie equally spaced on a circle of radius $r^{1/n}$.

8. What geometric object in the complex plane is described by $|z - z_0| = \rho$?

   A circle of radius $\rho$ centered at the point $z_0$. The inequality $|z - z_0| < \rho$ describes the open disk.

9. How do multiplication and division affect modulus and argument of complex numbers?

   $|z_1 z_2| = |z_1||z_2|$ and $\arg(z_1 z_2) = \arg z_1 + \arg z_2$; for division $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$ and $\arg\frac{z_1}{z_2} = \arg z_1 - \arg z_2$.

10. State the Fundamental Theorem of Algebra.

    Every non-constant polynomial $p(z)$ of degree $n$ with complex coefficients has exactly $n$ roots in $\mathbb{C}$, counted with multiplicity. Hence $\mathbb{C}$ is algebraically closed.

11. What does it mean for a power series $\sum_{n=0}^{\infty} a_n (z - z_0)^n$ to have radius of convergence $R$?

    The series converges absolutely for $|z - z_0| < R$ and diverges for $|z - z_0| > R$. $R$ is the radius of the disk of convergence; behavior on $|z-z_0|=R$ must be checked separately.

12. Give the Cauchy–Hadamard formula for the radius of convergence of $\sum a_n (z-z_0)^n$.

    $$\frac{1}{R} = \limsup_{n\to\infty} |a_n|^{1/n}.$$ Equivalently, if it exists, $R = \lim_{n\to\infty}\left|\frac{a_n}{a_{n+1}}\right|$.

13. What is the key analytic property of a power series within its disk of convergence?

    It defines a function that is analytic (holomorphic) there, and may be differentiated and integrated term by term; the differentiated series has the same radius of convergence $R$.

14. Define the complex exponential $e^{z}$ via its power series.

    $$e^{z} = \sum_{n=0}^{\infty} \frac{z^{n}}{n!},$$ convergent for all $z \in \mathbb{C}$ (entire function).

15. State Euler's formula and its consequence for $e^{x+iy}$.

    Euler: $e^{i\theta} = \cos\theta + i\sin\theta$. Hence $e^{x+iy} = e^{x}(\cos y + i\sin y)$, with $|e^{x+iy}| = e^{x}$.

16. Is the complex exponential function periodic? If so, give its period.

    Yes. $e^{z}$ is periodic with period $2\pi i$: $e^{z + 2\pi i} = e^{z}$ for all $z$. It is never zero.

17. Express $\cos z$ and $\sin z$ in terms of complex exponentials.

    $$\cos z = \frac{e^{iz} + e^{-iz}}{2}, \qquad \sin z = \frac{e^{iz} - e^{-iz}}{2i}.$$

18. Are the complex trigonometric functions $\sin z$ and $\cos z$ bounded? Explain.

    No. Unlike the real case, they are unbounded on $\mathbb{C}$. For example $|\sin z| \to \infty$ as $|\operatorname{Im} z| \to \infty$, since $\sin(iy) = i\sinh y$.

19. Define the complex hyperbolic functions $\cosh z$ and $\sinh z$.

    $$\cosh z = \frac{e^{z} + e^{-z}}{2}, \qquad \sinh z = \frac{e^{z} - e^{-z}}{2}.$$

20. State the relations connecting trigonometric and hyperbolic functions of a complex variable.

    $\cos(iz) = \cosh z$, $\sin(iz) = i\sinh z$, $\cosh(iz) = \cos z$, and $\sinh(iz) = i\sin z$.

21. Give the fundamental Pythagorean-type identities for complex $\sin,\cos,\sinh,\cosh$.

    $\sin^{2} z + \cos^{2} z = 1$ and $\cosh^{2} z - \sinh^{2} z = 1$, valid for all $z \in \mathbb{C}$.

22. What are transcendental functions in complex analysis, with examples?

    Functions not expressible as roots of a polynomial equation with rational-function coefficients — i.e. not algebraic. Examples: $e^{z}$, $\sin z$, $\cos z$, $\log z$, $\sinh z$. They are typically defined by power series.

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