GATE Physics Mathematical Physics Syllabus

How is the general solution of an inhomogeneous linear ODE structured?

$y = y_c + y_p$, where $y_c$ (complementary function) is the general solution of the associated homogeneous equation and $y_p$ is any particular solution of the full equation.

Write the standard form of a second-order linear ODE and its homogeneous version.

$\frac{d^{2}y}{dx^{2}}+P(x)\frac{dy}{dx}+Q(x)y = R(x)$; homogeneous when $R(x)=0$.

What is the Wronskian of two solutions $y_1,y_2$, and what does it test?

$W(y_1,y_2)=\begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix}=y_1 y_2' - y_2 y_1'$. If $W\neq 0$ on an interval, $y_1$ and $y_2$ are linearly independent there.

For a constant-coefficient ODE $ay''+by'+cy=0$, how do the roots of the auxiliary equation give the solution?

From $am^{2}+bm+c=0$: distinct real roots $m_1,m_2$ give $y=c_1e^{m_1x}+c_2e^{m_2x}$; a repeated root $m$ gives $y=(c_1+c_2x)e^{mx}$; complex roots $\alpha\pm i\beta$ give $y=e^{\alpha x}(c_1\cos\beta x + c_2\sin\beta x)$.

Classify a point $x_0$ of $y''+P(x)y'+Q(x)y=0$ as ordinary, regular singular, or irregular singular.

Ordinary if $P$ and $Q$ are analytic at $x_0$. Regular singular if $x_0$ is singular but $(x-x_0)P(x)$ and $(x-x_0)^{2}Q(x)$ are analytic. Otherwise it is an irregular singular point.

What method is used to solve a second-order ODE about a regular singular point?

The Frobenius method: assume $y=\sum_{k=0}^{\infty} a_k (x-x_0)^{k+s}$ with indicial exponent $s$ determined by the indicial equation, then find the recurrence for the coefficients $a_k$.

Write Legendre's differential equation and name its polynomial solutions.

$(1-x^{2})y'' - 2x y' + l(l+1)y = 0$. For integer $l$ the regular solutions are the Legendre polynomials $P_l(x)$, orthogonal on $[-1,1]$.

Write Bessel's differential equation of order $n$.

$x^{2}y'' + x y' + (x^{2}-n^{2})y = 0$, whose solutions are the Bessel functions $J_n(x)$ and $Y_n(x)$.

Write Hermite's differential equation and where its polynomial solutions appear in physics.

$y'' - 2x y' + 2n y = 0$, with Hermite polynomial solutions $H_n(x)$; they appear in the eigenfunctions of the quantum harmonic oscillator.

State the orthogonality relation for Legendre polynomials.

$\int_{-1}^{1} P_l(x)P_m(x)\,dx = \frac{2}{2l+1}\,\delta_{lm}$.

When is a complex function $f(z)$ called analytic (holomorphic) at a point?

When it is complex-differentiable not only at the point but throughout some neighbourhood of it; equivalently $f'(z)$ exists and is single-valued in that region.

State the Cauchy-Riemann conditions for $f(z)=u(x,y)+iv(x,y)$.

$\dfrac{\partial u}{\partial x}=\dfrac{\partial v}{\partial y}$ and $\dfrac{\partial u}{\partial y}=-\dfrac{\partial v}{\partial x}$. With continuous partial derivatives these are necessary and sufficient for analyticity.

What property do the real and imaginary parts of an analytic function satisfy individually?

They are harmonic: $\nabla^{2}u = \dfrac{\partial^{2}u}{\partial x^{2}}+\dfrac{\partial^{2}u}{\partial y^{2}}=0$ and likewise $\nabla^{2}v=0$. They are called conjugate harmonic functions.

Express the Cauchy-Riemann conditions in polar form for $f=u+iv$ with $z=re^{i\theta}$.

$\dfrac{\partial u}{\partial r}=\dfrac{1}{r}\dfrac{\partial v}{\partial \theta}$ and $\dfrac{1}{r}\dfrac{\partial u}{\partial \theta}=-\dfrac{\partial v}{\partial r}$.

State Cauchy's integral theorem.

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

State Cauchy's integral formula for $f$ and its $n$th derivative.

For $f$ analytic inside and on $C$ with $z_0$ enclosed: $f(z_0)=\dfrac{1}{2\pi i}\oint_C \dfrac{f(z)}{z-z_0}\,dz$, and $f^{(n)}(z_0)=\dfrac{n!}{2\pi i}\oint_C \dfrac{f(z)}{(z-z_0)^{n+1}}\,dz$.

What is a Laurent series and where is it valid?

An expansion $f(z)=\sum_{n=-\infty}^{\infty} a_n (z-z_0)^{n}$ valid in an annulus about $z_0$, including negative powers; it represents functions with singularities, unlike a Taylor series.

Classify the isolated singularities of a complex function.

Removable singularity (no negative powers in the Laurent series), pole of order $m$ (highest negative power is $(z-z_0)^{-m}$), and essential singularity (infinitely many negative-power terms).

How do you identify a pole of order $m$ at $z_0$?

The Laurent expansion has its most negative term $(z-z_0)^{-m}$, equivalently $(z-z_0)^{m} f(z)$ is analytic and nonzero at $z_0$. A pole of order 1 is called a simple pole.

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

It is the coefficient $a_{-1}$ of $(z-z_0)^{-1}$ in the Laurent expansion of $f$ about $z_0$, written $\operatorname{Res}_{z=z_0} f(z)=a_{-1}$.

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

$\operatorname{Res}_{z=z_0} f(z) = \lim_{z\to z_0} (z-z_0)f(z)$. If $f=p/q$ with a simple zero of $q$ at $z_0$, this equals $\dfrac{p(z_0)}{q'(z_0)}$.

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

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

State the residue theorem.

If $f$ is analytic inside and on a simple closed contour $C$ except at isolated singularities $z_k$ enclosed by $C$, then $\oint_C f(z)\,dz = 2\pi i \sum_k \operatorname{Res}_{z=z_k} f(z)$.

How is the residue theorem used to evaluate $\int_{-\infty}^{\infty} f(x)\,dx$ for a rational function?

Close the contour with a large semicircle in the upper half-plane; if $f\to 0$ fast enough the arc contributes nothing, giving $\int_{-\infty}^{\infty} f(x)\,dx = 2\pi i \sum \operatorname{Res}$ at the poles in the upper half-plane.

State Jordan's lemma and where it is applied.

For $\int_{-\infty}^{\infty} f(x)e^{iax}\,dx$ with $a>0$, if $f(z)\to 0$ uniformly as $|z|\to\infty$ in the upper half-plane, the contribution of the large semicircular arc vanishes, so the integral equals $2\pi i\sum\operatorname{Res}$ in the upper half-plane. It is used for Fourier-type integrals.

What is the residue at a simple pole on the real axis (principal value), and its contour contribution?

A simple pole $z_0$ on the contour contributes $i\pi\,\operatorname{Res}_{z=z_0} f$ (half the full $2\pi i$) via a small indenting semicircle, used in evaluating Cauchy principal-value integrals.

How are integrals of the form $\int_0^{2\pi} R(\cos\theta,\sin\theta)\,d\theta$ evaluated by residues?

Substitute $z=e^{i\theta}$, so $\cos\theta=\frac{1}{2}(z+z^{-1})$, $\sin\theta=\frac{1}{2i}(z-z^{-1})$, $d\theta=\dfrac{dz}{iz}$, turning it into a contour integral over the unit circle $|z|=1$ evaluated by the residue theorem.

What is the residue of $f(z)=\dfrac{e^{z}}{z^{2}}$ at $z=0$, and what type of singularity is it?

It is a pole of order 2. Using the order-$m$ formula, $\operatorname{Res}_{z=0}=\lim_{z\to0}\frac{d}{dz}\big[z^{2}\cdot \frac{e^{z}}{z^{2}}\big]=\frac{d}{dz}e^{z}\big|_{0}=1$.

State Liouville's theorem in complex analysis.

A function that is analytic (entire) and bounded on the whole complex plane must be constant.

What does the Sturm-Liouville form of an ODE guarantee for its eigenfunctions?

Writing the equation as $\dfrac{d}{dx}\!\left[p(x)\dfrac{dy}{dx}\right]+\big[q(x)+\lambda w(x)\big]y=0$ with suitable boundary conditions yields real eigenvalues and eigenfunctions orthogonal with respect to the weight $w(x)$: $\int y_m y_n\, w(x)\,dx=0$ for $m\neq n$.