CSIR NET Mathematical Science UNIT – 2 Flashcards

24 sample cards from the UNIT – 2 deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. State the sufficient condition (CR + smoothness) for analyticity.

   If $u,v$ have continuous first partial derivatives in a region and satisfy the Cauchy–Riemann equations there, then $f = u+iv$ is analytic in that region.

2. Express $f'(z)$ in terms of partial derivatives when $f = u+iv$ is analytic.

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

3. Give the Cauchy–Riemann equations in polar form.

   For $f = u(r,\theta) + iv(r,\theta)$: $$\frac{\partial u}{\partial r} = \frac{1}{r}\frac{\partial v}{\partial \theta}, \qquad \frac{1}{r}\frac{\partial u}{\partial \theta} = -\frac{\partial v}{\partial r}.$$

4. What is a harmonic function, and how does it relate to analytic functions?

   A real function $u$ is harmonic if $\Delta u = u_{xx} + u_{yy} = 0$. If $f = u+iv$ is analytic, both $u$ and $v$ are harmonic, and $v$ is the harmonic conjugate of $u$.

5. Define a contour integral $\int_{\gamma} f(z)\,dz$ via parametrization.

   If $\gamma$ is given by $z(t)$, $t \in [a,b]$, then $$\int_{\gamma} f(z)\,dz = \int_{a}^{b} f(z(t))\,z'(t)\,dt.$$

6. State the ML-inequality (estimation lemma) for contour integrals.

   If $|f(z)| \leq M$ on a contour $\gamma$ of length $L$, then $$\left|\int_{\gamma} f(z)\,dz\right| \leq M L.$$

7. Evaluate $\oint_{|z-z_0|=r} (z - z_0)^{n}\,dz$ for integer $n$.

   $$\oint (z-z_0)^{n}\,dz = \begin{cases} 2\pi i & n = -1,\\ 0 & n \neq -1.\end{cases}$$

8. State Cauchy's theorem (Cauchy–Goursat).

   If $f$ is analytic in a simply connected domain $D$, then for every closed contour $\gamma$ in $D$, $$\oint_{\gamma} f(z)\,dz = 0.$$

9. What does path independence of contour integrals require?

   If $f$ is analytic in a simply connected domain, the integral $\int_{z_1}^{z_2} f(z)\,dz$ is independent of the path joining $z_1$ to $z_2$; equivalently $f$ has an antiderivative $F$ with $\int = F(z_2) - F(z_1)$.

10. State Cauchy's integral formula.

    If $f$ is analytic inside and on a positively oriented simple closed contour $\gamma$, and $z_0$ is interior, then $$f(z_0) = \frac{1}{2\pi i}\oint_{\gamma} \frac{f(z)}{z - z_0}\,dz.$$

11. State Cauchy's integral formula for the $n$-th derivative.

    $$f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_{\gamma} \frac{f(z)}{(z - z_0)^{n+1}}\,dz.$$ Hence an analytic function is infinitely differentiable.

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

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

13. State Liouville's theorem.

    Every bounded entire function is constant. That is, if $f$ is analytic on all of $\mathbb{C}$ and $|f(z)| \leq M$ for all $z$, then $f$ is constant.

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

    If a non-constant polynomial $p$ had no zero, then $1/p$ would be a bounded entire function, hence constant by Liouville — a contradiction. So $p$ must have a root.

15. State the Maximum Modulus Principle.

    If $f$ is analytic and non-constant in a domain $D$, then $|f|$ has no local maximum in $D$. On a bounded region, $|f|$ attains its maximum only on the boundary.

16. State the Minimum Modulus Principle.

    If $f$ is analytic and non-constant in a domain $D$ and $f$ has no zeros in $D$, then $|f|$ attains its minimum only on the boundary (not at an interior point).

17. State the Mean Value Property of analytic functions.

    If $f$ is analytic on and inside $|z - z_0| = r$, then $$f(z_0) = \frac{1}{2\pi}\int_{0}^{2\pi} f(z_0 + re^{i\theta})\,d\theta,$$ the average of $f$ over the circle.

18. State Schwarz's Lemma.

    If $f$ is analytic on the unit disk $|z|<1$ with $f(0)=0$ and $|f(z)| \leq 1$, then $|f(z)| \leq |z|$ for all $z$, and $|f'(0)| \leq 1$.

19. State the equality (rigidity) case of Schwarz's Lemma.

    If in Schwarz's Lemma $|f(z_0)| = |z_0|$ for some $z_0 \neq 0$, or $|f'(0)| = 1$, then $f(z) = e^{i\alpha} z$ for some real constant $\alpha$ (a rotation).

20. State the Open Mapping Theorem.

    A non-constant analytic function on a domain $D$ maps open sets to open sets; i.e. the image of every open subset of $D$ is open.

21. How does the Open Mapping Theorem relate to the Maximum Modulus Principle?

    Since a non-constant analytic $f$ sends open sets to open sets, $f(z_0)$ is an interior point of the image, so $|f|$ cannot attain a local maximum at an interior $z_0$ — giving the Maximum Modulus Principle.

22. State Taylor's theorem for analytic functions.

    If $f$ is analytic on a disk $|z - z_0| < R$, then $$f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!}(z - z_0)^{n}$$ converging for all $z$ in that disk.

23. What determines the radius of convergence of the Taylor series of an analytic function about $z_0$?

    It equals the distance from $z_0$ to the nearest singularity of $f$. The Taylor series converges in the largest open disk centered at $z_0$ containing no singularity.

24. Give the Taylor (Maclaurin) series of $e^{z}$, $\sin z$, and $\cos z$ about $0$.

    $$e^{z} = \sum_{n=0}^{\infty}\frac{z^{n}}{n!},\quad \sin z = \sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{(2n+1)!},\quad \cos z = \sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n}}{(2n)!}.$$ All converge for every $z \in \mathbb{C}$.

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