CSIR NET Mathematical Science UNIT – 2 Syllabus

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 $z \in \mathbb{C}$.

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.

Define the principal complex logarithm $\operatorname{Log} z$.

For $z \neq 0$, $$\operatorname{Log} z = \ln|z| + i\,\operatorname{Arg} z,$$ where $\operatorname{Arg} z \in (-\pi, \pi]$ is the principal argument. It is multi-valued in general: $\log z = \ln|z| + i(\arg z + 2\pi k)$.

Define what it means for $f$ to be analytic (holomorphic) at a point $z_0$.

$f$ is analytic at $z_0$ if it is complex-differentiable at every point in some open neighborhood of $z_0$, i.e. the limit $f'(z) = \lim_{h\to 0}\frac{f(z+h)-f(z)}{h}$ exists throughout that neighborhood.

What is an entire function? Give examples.

A function that is analytic on all of $\mathbb{C}$. Examples: polynomials, $e^{z}$, $\sin z$, $\cos z$, $\sinh z$, $\cosh z$.

State the Cauchy–Riemann equations for $f = u + iv$.

With $u = u(x,y)$, $v = v(x,y)$: $$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.$$

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.

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

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

Give the Cauchy–Riemann equations in polar form.

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

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 $u$.

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

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

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

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

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

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

State Cauchy's theorem (Cauchy–Goursat).

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

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 $\int = F(z_2) - F(z_1)$.

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 $$f(z_0) = \frac{1}{2\pi i}\oint_{\gamma} \frac{f(z)}{z - z_0}\,dz.$$

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.

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 $$|f^{(n)}(z_0)| \leq \frac{n!\,M}{R^{n}}.$$

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.

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.

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.

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).

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.

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 $|f'(0)| \leq 1$.

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).

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.

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.

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.

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.

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 $z \in \mathbb{C}$.