GATE Mathematics Complex Analysis Flashcards

24 sample cards from the Complex Analysis deck

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

1. How is the radius of convergence $R$ of $\sum a_n (z-z_0)^n$ found via the root test?

   By the Cauchy–Hadamard formula: $$\frac{1}{R}=\limsup_{n\to\infty}|a_n|^{1/n}.$$

2. Give the ratio-test formula for the radius of convergence of $\sum a_n (z-z_0)^n$.

   $$R=\lim_{n\to\infty}\left|\frac{a_n}{a_{n+1}}\right|,$$ provided the limit exists.

3. What is the relationship between the radius of convergence of a Taylor series and the nearest singularity?

   The radius of convergence equals the distance from the center $z_0$ to the nearest singularity of $f$ in the complex plane.

4. State the Laurent series expansion of $f$ in an annulus.

   If $f$ is analytic in the annulus $r<|z-z_0|<R$, then $$f(z)=\sum_{n=-\infty}^{\infty} a_n (z-z_0)^{n},\quad a_n=\frac{1}{2\pi i}\oint_C \frac{f(z)}{(z-z_0)^{n+1}}\,dz.$$

5. In a Laurent series, what are the principal part and the analytic part?

   The principal part is $\sum_{n=1}^{\infty} a_{-n}(z-z_0)^{-n}$ (negative powers); the analytic (regular) part is $\sum_{n=0}^{\infty} a_n (z-z_0)^{n}$ (non-negative powers).

6. How does the structure of the principal part of a Laurent series classify an isolated singularity?

   Removable: no negative powers. Pole of order $m$: finitely many negative powers, lowest being $(z-z_0)^{-m}$. Essential singularity: infinitely many negative powers.

7. Define the residue of $f$ at an isolated singularity $z_0$.

   The residue $\operatorname{Res}_{z=z_0} f$ is the coefficient $a_{-1}$ of $(z-z_0)^{-1}$ in the Laurent expansion of $f$ about $z_0$. Equivalently $a_{-1}=\frac{1}{2\pi i}\oint_C f\,dz$.

8. Give the formula for the residue at a simple pole $z_0$.

   $$\operatorname{Res}_{z=z_0} f=\lim_{z\to z_0}(z-z_0)\,f(z).$$ If $f=p/q$ with simple pole, $\operatorname{Res}=\frac{p(z_0)}{q'(z_0)}$.

9. Give the formula for the residue at a pole of order $m$.

   $$\operatorname{Res}_{z=z_0} f=\frac{1}{(m-1)!}\lim_{z\to z_0}\frac{d^{\,m-1}}{dz^{\,m-1}}\big[(z-z_0)^{m} f(z)\big].$$

10. State the Residue Theorem.

    If $f$ is analytic inside and on a positively oriented simple closed contour $C$ except for isolated singularities $z_1,\dots,z_k$ inside, then $$\oint_C f(z)\,dz=2\pi i\sum_{j=1}^{k}\operatorname{Res}_{z=z_j} f.$$

11. How is a real integral $\int_0^{2\pi} R(\cos\theta,\sin\theta)\,d\theta$ converted to a contour integral?

    Substitute $z=e^{i\theta}$, so $\cos\theta=\frac{1}{2}(z+z^{-1})$, $\sin\theta=\frac{1}{2i}(z-z^{-1})$, $d\theta=\frac{dz}{iz}$, turning it into $\oint_{|z|=1}$ evaluated by residues inside the unit circle.

12. Describe the contour method to evaluate $\int_{-\infty}^{\infty} \dfrac{P(x)}{Q(x)}\,dx$ for rational functions.

    Close the contour with a large semicircle in the upper half-plane. If $\deg Q\ge \deg P+2$, the arc contribution vanishes, so the integral equals $2\pi i$ times the sum of residues at poles in the upper half-plane.

13. State Jordan's Lemma and its use.

    For $m>0$, if $f(z)\to 0$ uniformly as $|z|\to\infty$ on the upper semicircle $C_R$, then $\int_{C_R} f(z)e^{imz}\,dz\to 0$. It justifies dropping the arc when evaluating $\int_{-\infty}^{\infty} f(x)e^{imx}\,dx$ by residues.

14. How are integrals of the form $\int_{-\infty}^{\infty} f(x)\cos(mx)\,dx$ or $\int f(x)\sin(mx)\,dx$ evaluated?

    Consider $\int_{-\infty}^{\infty} f(x)e^{imx}\,dx=2\pi i\sum(\text{residues in upper half-plane})$, then take the real part for $\cos$ and the imaginary part for $\sin$.

15. What is the contribution of a simple pole lying on the real axis (contour indentation)?

    Indenting around a simple pole $x_0$ on the real axis with a small semicircle contributes $\pm i\pi\,\operatorname{Res}_{z=x_0} f$ (half the full $2\pi i$ residue), the sign depending on orientation; the integral is taken as a Cauchy principal value.

16. Classify the singularity of $f(z)=\dfrac{\sin z}{z}$ at $z=0$.

    Removable singularity. The Laurent series has no negative powers ($\frac{\sin z}{z}=1-\frac{z^2}{3!}+\cdots$), and $f$ can be defined as $1$ at $z=0$ to make it analytic.

17. Classify the singularity of $e^{1/z}$ at $z=0$ and state the relevant theorem.

    Essential singularity (Laurent series $\sum_{n=0}^\infty \frac{1}{n!\,z^n}$ has infinitely many negative powers). By the Casorati–Weierstrass / Picard theorem, $f$ comes arbitrarily close to every complex value near $z=0$.

18. Compare a pole and an essential singularity in terms of $\lim_{z\to z_0}|f(z)|$.

    At a pole, $|f(z)|\to\infty$ as $z\to z_0$. At an essential singularity, the limit does not exist and $f$ takes values arbitrarily close to every complex number (Casorati–Weierstrass).

19. State the Cauchy product / uniqueness fact: how do the Taylor and Laurent coefficients relate when $f$ is analytic at $z_0$?

    If $f$ is analytic at $z_0$, the Laurent series reduces to the Taylor series, so $a_n=\frac{f^{(n)}(z_0)}{n!}$ for $n\ge0$ and $a_n=0$ for $n<0$. The expansion is unique.

20. What does it mean for a series $\sum a_n(z-z_0)^n$ when $z$ is exactly on the circle $|z-z_0|=R$?

    The behaviour on the circle of convergence is indeterminate by the radius test alone: the series may converge at some boundary points and diverge at others; it must be examined separately.

21. Give the Taylor (Maclaurin) series of $e^z$, $\sin z$, and $\cos z$ with their radii of convergence.

    $e^z=\sum_{n=0}^\infty \frac{z^n}{n!}$, $\sin z=\sum_{n=0}^\infty \frac{(-1)^n z^{2n+1}}{(2n+1)!}$, $\cos z=\sum_{n=0}^\infty \frac{(-1)^n z^{2n}}{(2n)!}$. Each has $R=\infty$ (entire).

22. Give the geometric series and its radius of convergence.

    $$\frac{1}{1-z}=\sum_{n=0}^{\infty} z^{n},\quad |z|<1,\ R=1.$$ It diverges for $|z|\ge1$; the singularity at $z=1$ sets $R$.

23. Define the winding number (index) of a closed contour $C$ about a point $z_0$.

    $$n(C,z_0)=\frac{1}{2\pi i}\oint_C \frac{dz}{z-z_0},$$ an integer counting the net number of times $C$ winds counterclockwise around $z_0$.

24. State the Argument Principle.

    If $f$ is meromorphic inside and on a simple closed contour $C$ (no zeros/poles on $C$), then $$\frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}\,dz=Z-P,$$ where $Z$ and $P$ are the numbers of zeros and poles inside $C$, counted with multiplicity.

What this deck covers

Complex Analysis flashcards FAQ