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GATE Physics Mathematical Physics Flashcards
50 question-and-answer cards covering Mathematical Physics as it is examined in GATE Physics. 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.
24 sample cards from the Mathematical Physics deck
Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.
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 $m\neq n$.
What this deck covers
The Mathematical Physics deck follows the GATE Physics Mathematical Physics syllabus — 1 chapters and 7 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 50.0 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 172 characters, which is long enough to carry the reasoning and short enough to say out loud.
A deck like this earns its keep on the second and third pass. Read the syllabus first so you know the shape of the subject, then use the cards to find the specific facts that have not stuck.
Mathematical Physics flashcards FAQ
How many Mathematical Physics flashcards are in this GATE Physics deck?
50 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.
Are these GATE Physics flashcards free?
Yes. The preview here is free to read with no signup, and the full 50-card deck is free inside the Examius app.
What do the Mathematical Physics cards cover?
They follow the GATE Physics Mathematical Physics syllabus — 1 chapters and 7 topics — so the questions track what is actually examinable.
How should I use these flashcards?
Read the syllabus first so you know the shape of the subject, then drill the deck. Examius schedules each card with spaced repetition, so cards you keep missing come back sooner and ones you know drift further apart.