GATE Mathematics Complex Analysis Syllabus

Complex Analysis syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Complex Analysis in GATE Mathematics, not a summary of it.

1. 1

   Functions of a Complex Variable

   3 topics

   • Continuity
   • Differentiability
   • Analytic Functions
2. 2

   Complex Integration

   5 topics

   • Cauchy’s Integral Theorem
   • Cauchy’s Integral Formula
   • Liouville’s Theorem
   • Maximum Modulus Principle
   • Morera’s Theorem
3. 3

   Zeros and Singularities

   overview

   Examined as a single unit within Complex Analysis — no further topic split in the official outline.

4. 4

   Power Series

   3 topics

   • Radius of Convergence
   • Taylor’s Series
   • Laurent’s Series
5. 5

   Residue Theorem and Applications

   1 topic

   • Evaluating Real Integrals
6. 6

   Rouche’s Theorem

   overview

   Examined as a single unit within Complex Analysis — no further topic split in the official outline.

7. 7

   Argument Principle

   overview

   Examined as a single unit within Complex Analysis — no further topic split in the official outline.

8. 8

   Schwarz Lemma

   overview

   Examined as a single unit within Complex Analysis — no further topic split in the official outline.

9. 9

   Conformal Mappings

   overview

   Examined as a single unit within Complex Analysis — no further topic split in the official outline.

10. 10

    Mobius Transformations

    overview

    Examined as a single unit within Complex Analysis — no further topic split in the official outline.

Complex Analysis flashcards for GATE Mathematics

22 of 50 cards from the Complex Analysis deck — real questions with worked answers.

1. State the $\epsilon$-$\delta$ definition of continuity of a complex function $f$ at a point $z_0$.

   $f$ is continuous at $z_0$ if for every $\epsilon>0$ there exists $\delta>0$ such that $|z-z_0|<\delta \implies |f(z)-f(z_0)|<\epsilon$. Equivalently, $\lim_{z\to z_0} f(z)=f(z_0)$.

2. If $f(z)=u(x,y)+i\,v(x,y)$, what does continuity of $f$ require of $u$ and $v$?

   $f$ is continuous at $z_0=x_0+iy_0$ if and only if both real-valued functions $u(x,y)$ and $v(x,y)$ are continuous at $(x_0,y_0)$.

3. Define the complex derivative of $f$ at $z_0$.

   $$f'(z_0)=\lim_{\Delta z\to 0}\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z},$$ provided the limit exists independently of the direction in which $\Delta z\to 0$.

4. State the Cauchy–Riemann equations in Cartesian form.

   For $f=u+iv$, $$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\qquad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.$$

5. State the sufficient condition for $f=u+iv$ to be differentiable (analytic) at a point.

   If $u,v$ have continuous first-order partial derivatives in a neighborhood of $z_0$ and satisfy the Cauchy–Riemann equations there, then $f$ is differentiable at $z_0$.

6. Give the Cauchy–Riemann equations in polar coordinates.

   $$\frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial \theta},\qquad \frac{1}{r}\frac{\partial u}{\partial \theta}=-\frac{\partial v}{\partial r}.$$

7. Express $f'(z)$ in terms of partial derivatives of $u$ and $v$.

   $$f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}.$$

8. Define an analytic (holomorphic) function on a domain.

   $f$ is analytic on an open set (domain) $D$ if it is complex-differentiable at every point of $D$. Analyticity at a point means differentiability in some neighborhood of that point.

9. What is an entire function? Give two examples.

   An entire function is analytic on the whole complex plane $\mathbb{C}$. Examples: $e^{z}$, $\sin z$, $\cos z$, and any polynomial.

10. Define a harmonic function and state its relation to analytic functions.

    A real function $\phi$ is harmonic if it satisfies Laplace's equation $\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}=0$. The real and imaginary parts $u,v$ of an analytic function are harmonic; $v$ is the harmonic conjugate of $u$.

11. Is differentiability at a single point enough for analyticity? Illustrate with $f(z)=|z|^2$.

    No. Analyticity requires differentiability in a neighborhood. $f(z)=|z|^2$ is differentiable only at $z=0$ (CR equations hold only there) but is nowhere analytic.

12. State Cauchy's Integral Theorem (Cauchy–Goursat theorem).

    If $f$ is analytic on and inside a simple closed contour $C$ within a simply connected domain, then $$\oint_C f(z)\,dz=0.$$

13. What is the consequence of Cauchy's theorem for path independence of integrals?

    If $f$ is analytic in a simply connected domain, the contour integral $\int_{z_1}^{z_2} f(z)\,dz$ depends only on the endpoints, not the path; $f$ has an analytic antiderivative there.

14. State the deformation of contours principle.

    If $f$ is analytic in the region between two simple closed contours $C_1$ and $C_2$ (with $C_2$ inside $C_1$), then $\oint_{C_1} f\,dz=\oint_{C_2} f\,dz$; a contour may be continuously deformed without crossing singularities.

15. State Cauchy's Integral Formula for $f(z_0)$.

    If $f$ is analytic on and inside a simple closed contour $C$ and $z_0$ lies inside $C$, then $$f(z_0)=\frac{1}{2\pi i}\oint_C \frac{f(z)}{z-z_0}\,dz.$$

16. State Cauchy's Integral Formula for the $n$-th derivative.

    $$f^{(n)}(z_0)=\frac{n!}{2\pi i}\oint_C \frac{f(z)}{(z-z_0)^{n+1}}\,dz,\quad n=0,1,2,\dots$$

17. What key fact about analytic functions follows from Cauchy's integral formula for derivatives?

    If $f$ is analytic in a domain, it is infinitely differentiable there; all its derivatives $f',f'',\dots$ exist and are themselves analytic.

18. State Cauchy's inequality (estimate) for derivatives.

    If $f$ is analytic inside and on a circle $|z-z_0|=R$ with $|f(z)|\le M$ on it, then $$|f^{(n)}(z_0)|\le \frac{n!\,M}{R^{n}}.$$

19. Evaluate $\oint_C \dfrac{1}{z-a}\,dz$ where $C$ is a positively oriented simple closed contour enclosing $a$.

    $$\oint_C \frac{1}{z-a}\,dz=2\pi i.$$ If $a$ lies outside $C$, the integral is $0$.

20. State Liouville's Theorem.

    If $f$ is entire (analytic on all of $\mathbb{C}$) and bounded, i.e. $|f(z)|\le M$ for all $z$, then $f$ is constant.

21. How does Liouville's Theorem prove the Fundamental Theorem of Algebra?

    If a non-constant polynomial $p(z)$ had no zero, then $1/p(z)$ would be entire and bounded, hence constant by Liouville — a contradiction. Thus every non-constant polynomial has at least one root in $\mathbb{C}$.

22. State the Maximum Modulus Principle.

    If $f$ is analytic and non-constant in a bounded domain $D$, then $|f(z)|$ attains its maximum only on the boundary $\partial D$, never at an interior point of $D$.

See more Complex Analysis flashcards →

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