Mathematics for Chemists: Fourier Series, PDEs, and Power Series Solutions

Comprehensive practice flashcards covering Fourier series, Fourier transforms, separation of variables for partial differential equations, and power series solutions of ordinary differential equations.

What defines a function $f(x)$ as periodic with period $P$?

It is invariant under translations $x \mapsto x + P$, satisfying $f(x) = f(x + P)$ for all real $x$.

What is the fundamental period of a periodic function $f(x)$?

The smallest positive constant $P$ for which $f(x) = f(x + P)$ holds.

How does scaling the argument of a periodic function $f(x)$ by a factor $k > 0$ affect its fundamental period, as depicted in this diagram?

The scaled function $f(kx)$ has a period scaled by a factor of $\frac{1}{k}$, giving a new period of $\frac{P}{k}$.

Under what condition is the sum of two periodic functions $f(x)$ with period $P_1$ and $g(x)$ with period $P_2$ also periodic?

When their periods are commensurate, such that there exist positive integers $m$ and $n$ where $P = m P_1 = n P_2$ is a least common multiple.

What is the mathematical condition for a function $f(x)$ to be classified as even?

$f(-x) = f(x)$ for all $x$ in its symmetric domain.

What geometric symmetry is exhibited by the graph of an odd function?

Rotational symmetry about the origin through an angle of $\pi$.

Why must the graph of an odd function pass through the origin if defined at $x = 0$?

Setting $x = 0$ in $f(-x) = -f(x)$ gives $f(0) = -f(0)$, which requires $2f(0) = 0$, hence $f(0) = 0$.

How can any function $f(x)$ defined on a symmetric interval be decomposed into even and odd components?

$f(x) = f_{\text{even}}(x) + f_{\text{odd}}(x)$, where $f_{\text{even}}(x) = \frac{1}{2}(f(x) + f(-x))$ and $f_{\text{odd}}(x) = \frac{1}{2}(f(x) - f(-x))$.

What is the parity of the product of an even function and an odd function?

The product is an odd function.

What is the definite integral of an odd function $f(x)$ over a symmetric interval $[-L, L]$?

$\int_{-L}^L f(x)\,dx = 0$.

What is the definite integral of an even function $f(x)$ over a symmetric interval $[-L, L]$?

$\int_{-L}^L f(x)\,dx = 2\int_0^L f(x)\,dx$.

What is the origin of the terms 'even' and 'odd' for functions?

Their convergent Maclaurin series contain exclusively even powers of $x$ (like $\cos(x)$) or exclusively odd powers of $x$ (like $\sin(x)$).

What orthogonality relation is satisfied by $\cos(mx)$ and $\cos(nx)$ on the interval $[-\pi, \pi]$ for positive integers $m$ and $n$?

$\frac{1}{\pi}\int_{-\pi}^\pi \cos(mx)\cos(nx)\,dx = \delta_{mn}$, where $\delta_{mn}$ is the Kronecker delta.

What is the value of $\int_{-\pi}^\pi \cos(mx)\sin(nx)\,dx$ for all positive integers $m$ and $n$?

$0$, because the integrand is an odd function integrated over a symmetric interval.

How is the trigonometric Fourier series defined for a $P$-periodic function $f(x)$?

$f(x) = A_0 + \sum_{n=1}^\infty A_n \cos\left(\frac{2n\pi x}{P}\right) + \sum_{n=1}^\infty B_n \sin\left(\frac{2n\pi x}{P}\right)$.

What is the integral formula for the Fourier coefficient $A_0$ of a $P$-periodic function $f(x)$?

$A_0 = \frac{1}{P}\int_{-P/2}^{P/2} f(x)\,dx$.

What is the formula for the Fourier coefficients $A_n$ ($n \ge 1$) of a $P$-periodic function $f(x)$?

$A_n = \frac{2}{P}\int_{-P/2}^{P/2} \cos(k_n x)f(x)\,dx$, where $k_n = \frac{2n\pi}{P}$.

What is the formula for the Fourier coefficients $B_n$ ($n \ge 1$) of a $P$-periodic function $f(x)$?

$B_n = \frac{2}{P}\int_{-P/2}^{P/2} \sin(k_n x)f(x)\,dx$, where $k_n = \frac{2n\pi}{P}$.

According to Dirichlet's theorem, to what value does the Fourier series converge at a point of discontinuity of $f(x)$?

To the mean of the left-hand and right-hand limits of the function at that point.

Which Fourier coefficients vanish if a periodic function $f(x)$ is strictly even?

$B_n = 0$ for all $n \ge 1$.

Which Fourier coefficients vanish if a periodic function $f(x)$ is strictly odd?

$A_0 = 0$ and $A_n = 0$ for all $n \ge 1$.

In signal analysis applications, what physical interpretation is given to the Fourier coefficient $A_0$?

The direct current (dc) offset, which is the mean value of the function over one period about which the alternating current (ac) signal is centred.

How is the complex exponential Fourier series defined for a $P$-periodic function $f(x)$?

$f(x) = \sum_{n=-\infty}^\infty C_n e^{i k_n x}$, where $k_n = \frac{2n\pi}{P}$.

What is the formula for the complex Fourier coefficients $C_n$ of a $P$-periodic function $f(x)$?

$C_n = \frac{1}{P}\int_{-P/2}^{P/2} e^{-i k_n x}f(x)\,dx$ for $n = 0, \pm 1, \pm 2, \dots$.

How is the odd periodic extension $f_{\text{odd}}(x)$ constructed for a function defined on $0 < x < P$?

$f_{\text{odd}}(x) = f(x)$ on $0 < x < P$, $0$ at $x = 0$ and $x = P$, and $-f(-x)$ on $-P < x < 0$, satisfying $f_{\text{odd}}(x + 2P) = f_{\text{odd}}(x)$.

For a plucked guitar string released from rest ($u_t(x,0) = 0$), why do the coefficients $B_n$ vanish in its normal-mode expansion?

The initial velocity condition gives $u_t(x,0) = \sum_{n=1}^\infty \omega_n B_n \sin(k_n x) = 0$, which by the uniqueness of Fourier sine series requires $B_n = 0$ for all $n$.

How does a discrete Fourier spectrum transition to a continuous Fourier spectrum as the period $P \to \infty$?

The discrete wavenumbers $k_n$ become infinitesimally spaced to form a continuous variable $k$, turning the Fourier summation into an integral.

What are the standard definitions of the continuous Fourier transform $\tilde{f}(k)$ and inverse Fourier transform $f(x)$?

$\tilde{f}(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(x)e^{-ikx}\,dx$ and $f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \tilde{f}(k)e^{ikx}\,dk$.

What is the Fourier transform of the two-sided exponential decay function $f(x) = e^{-|x|}$?

The Lorentzian function $\tilde{f}(k) = \sqrt{\frac{2}{\pi}}\frac{1}{1+k^2}$.

What is the Fourier transform of a symmetric top-hat function $\chi(x) = 1$ for $|x| \le 1$ and $0$ for $|x| > 1$?

$\tilde{\chi}(k) = \sqrt{\frac{2}{\pi}}\operatorname{sinc}(k)$, where $\operatorname{sinc}(k) = \frac{\sin(k)}{k}$ for $k \ne 0$ and $1$ for $k = 0$.

What is the Fourier transform of the Gaussian function $f(x) = e^{-x^2/2}$?

$\tilde{f}(k) = e^{-k^2/2}$.

What is the scaling rule for the Fourier transform of $f(x/\alpha)$ for a non-zero constant $\alpha$?

$\mathcal{F}[f(x/\alpha)] = |\alpha|\tilde{f}(\alpha k)$.

State the derivative rule for the Fourier transform of the $n$-th derivative $\frac{d^n f}{dx^n}$.

$\mathcal{F}\left[\frac{d^n f}{dx^n}\right] = (ik)^n \tilde{f}(k)$.

What are the three principal advantages of a Fourier transform spectrometer compared to a scanning dispersive instrument?

Fellgett's advantage (multiplex), Jacquinot's advantage (throughput), and Connes' advantage (wavelength accuracy).

How does the general solution of a partial differential equation differ fundamentally from that of an ordinary differential equation?

The general solution of an ODE depends on arbitrary integration constants, whereas the general solution of a PDE depends on arbitrary functions.

What is the general solution of the PDE $\frac{\partial^2 f}{\partial x\partial t} = (1+x)t$?

$f(x,t) = \frac{1}{4}(1+x)^2 t^2 + u(t) + v(x)$, where $u(t)$ and $v(x)$ are arbitrary functions.

Why does heat flow in a 1D bar require one initial condition and two boundary conditions?

Because the governing heat equation is first order in time (requiring one initial condition) and second order in space (requiring two boundary conditions).

What is the standard linear partial differential equation for diffusion with diffusivity $D$?

$D\nabla^2\psi = \frac{\partial\psi}{\partial t}$.

What is the standard linear partial differential equation describing classical waves propagating at speed $c$?

$c^2\nabla^2\psi = \frac{\partial^2\psi}{\partial t^2}$.

Write the time-dependent Schrödinger equation for a quantum particle of mass $m$ in a potential $U(x,y,z)$.

$-\frac{\hbar^2}{2m}\nabla^2\psi + U\psi = i\hbar\frac{\partial\psi}{\partial t}$.

What is Laplace's equation for a potential $\psi$, and what name is given to its solutions?

$\nabla^2\psi = 0$, and its solutions are called harmonic functions.

Write Poisson's equation relating electrostatic potential $\psi$ to charge density $\rho$.

$\nabla^2\psi = -\frac{\rho}{\epsilon_0}$.

State Maxwell's equations in free space without charge or current sources.

$\nabla\cdot\mathbf{E} = 0$, $\nabla\cdot\mathbf{B} = 0$, $\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}$, and $\nabla\times\mathbf{B} = \epsilon_0\mu_0\frac{\partial\mathbf{E}}{\partial t}$.

What is the Korteweg–de Vries (KdV) equation, and what physical wave phenomenon does it predict?

$\frac{\partial^3\psi}{\partial x^3} - 6\psi\frac{\partial\psi}{\partial x} + \frac{\partial\psi}{\partial t} = 0$; it models shallow water disturbances and predicts solitary waves (solitons).

Write the Navier–Stokes equation governing the motion of a viscous fluid.

$\frac{\partial\mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u} = \frac{1}{\rho}\nabla\cdot\boldsymbol{\sigma} + \mathbf{F}$.

What major mathematical open question concerning the Navier–Stokes equations is one of the Clay Millennium Problems?

Whether smooth, physically reasonable solutions always exist in three space dimensions.

What is Step 1 in the separation of variables method for solving a PDE?

Assume the separation ansatz: the solution factors into a product of single-variable functions, $f(x_1, \dots, x_n) = F_1(x_1)F_2(x_2)\dots F_n(x_n)$.

What is Step 2 in the separation of variables method for solving a PDE?

Set the separated single-variable expressions equal to a common separation constant $\mu$, because independent variables can only cancel to zero if both expressions are constant.

What is Step 3 in the separation of variables method for solving a PDE?

Determine the allowed values of the separation constant $\mu$ by imposing boundary conditions and rejecting trivial solutions that vanish identically.

For a quantum particle in a circular box of radius $a$, what boundary condition is imposed at the edge $r = a$?

$\psi(a, \theta) = 0$, because the potential is infinite outside the disk.

In the circular box problem, why can the separation constant $\mu$ in $\Theta'' + \mu\Theta = 0$ not be negative?

If $\mu = -q^2 < 0$, the solution is a linear combination of real exponentials $e^{\pm q\theta}$, which cannot satisfy $\Theta(\theta+2\pi) = \Theta(\theta)$ without setting all coefficients to zero.

What discrete values must the separation constant $\mu$ take to satisfy periodic boundary conditions in the circular box angular equation?

$\mu = N^2$ for integers $N = 0, \pm 1, \pm 2, \dots$.

What differential equation governs the radial wavefunction $R(X)$ for a quantum particle in a circular box?

Bessel's equation: $X^2 \frac{d^2 R}{dX^2} + X\frac{dR}{dX} + (X^2 - N^2)R = 0$.

What is the quantized energy spectrum $E_{N,k}$ for a particle of mass $m$ in a circular box of radius $a$?

$E_{N,k} = \left(\frac{\hbar^2}{2ma^2}\right) X_{|N|,k}^2$, where $X_{|N|,k}$ is the $k$-th zero of the Bessel function $J_{|N|}(X)$.

Why is the order of the Bessel function in the circular box wavefunction written as $|N|$ rather than $N$?

Because the physical solution regular at the origin is indexed by non-negative orders, and integer Bessel functions satisfy $J_{-N}(x) = (-1)^N J_N(x)$.

What ODE governs the angular factor $\Theta(\theta)$ when separating Laplace's equation in axisymmetric spherical polar coordinates?

$\frac{d^2\Theta}{d\theta^2} + \cot(\theta)\frac{d\Theta}{d\theta} + \mu\Theta = 0$.

What change of variable converts the axisymmetric spherical angular equation into Legendre's differential equation?

$\chi = \cos(\theta)$.

What class of ODE is the radial equation $r^2 R'' + 2r R' - N(N+1)R = 0$, and how is it solved?

It is an Euler–Cauchy (equidimensional) equation, transformed into a constant-coefficient ODE via the substitution $X = \ln(r)$.

What is the family of general solutions for the radial function $R_N(r)$ in axisymmetric spherical potential problems?

$R_N(r) = A_N r^N + B_N r^{-(N+1)}$.

Write the general series solution for an axisymmetric electrostatic potential $\psi(r,\theta)$ in spherical coordinates.

$\psi(r,\theta) = \sum_{N=0}^\infty \left(A_N r^N + \frac{B_N}{r^{N+1}}\right) P_N(\cos(\theta))$.

In 1D heat diffusion with zero-temperature boundary conditions, why must the spatial separation constant be negative ($-\mu$ with $\mu > 0$)?

To generate oscillatory solutions that can satisfy zero boundary conditions at both ends without vanishing identically.

What are the spatial eigenfunctions $X_N(x)$ for 1D heat diffusion in a rod of length $L$ with zero boundary conditions?

$X_N(x) = \sin\left(\frac{N\pi x}{L}\right)$ for $N = 1, 2, 3, \dots$.

What is the temporal decay factor $T_N(t)$ associated with the $N$-th mode in 1D heat diffusion?

$T_N(t) = \exp\left(-\frac{N^2\pi^2\kappa t}{L^2}\right)$.

What orthogonality relation is used to extract the mode coefficients $A_N$ from an initial temperature profile $g(x)$ in a rod of length $L$?

$\int_0^L \sin\left(\frac{M\pi x}{L}\right)\sin\left(\frac{N\pi x}{L}\right)\,dx = \frac{L}{2}\delta_{MN}$.

Why can separation of variables not be applied directly to a capillary diffusion problem with boundary conditions $\psi(0,t) = 1$ and $\psi(L,t) = 0$?

Because the boundary condition at $x = 0$ is nonhomogeneous.

How is a diffusion problem with nonhomogeneous boundary conditions converted into a tractable problem for separation of variables?

By expressing the solution as the sum of a steady-state equilibrium profile and a transient profile, $\psi(x,t) = \psi_{\text{eq}}(x) + \psi_0(x,t)$, where $\psi_0$ satisfies homogeneous boundary conditions.

What physical transport problem with nonhomogeneous boundary conditions is illustrated in this diagram?

Solute diffusion along a capillary tube connecting two well-stirred reservoirs (brine with concentration $1$ and pure water with concentration $0$).

When is a function $f(x)$ said to be analytic at a point $x = x_0$?

When its Taylor series expanded about $x_0$ is convergent.

What is an order-$s$ recurrence relation for a sequence $a_n$?

A relation of the form $F_s(a_n, a_{n+1}, \dots, a_{n+s}) = 0$ that relates $s + 1$ consecutive terms of the sequence.

What makes a recurrence relation linear and homogeneous?

It is a linear function of the sequence terms, and it is satisfied by the zero sequence ($a_n = 0$ for all $n$).

What recurrence relation is obtained when substituting a Maclaurin series into $\frac{df}{dx} = -kf$?

$a_{n+1} = -\frac{k}{n+1}a_n$.

What is the closed form of the sequence coefficients solving $\frac{df}{dx} = -kf$ via power series?

$a_n = \frac{(-k)^n}{n!}a_0$.

What is the standard general form of a second-order, linear, homogeneous ODE with variable coefficients?

$\frac{d^2 f}{dx^2} + p(x)\frac{df}{dx} + q(x)f = 0$.

Write the one-dimensional time-independent Schrödinger equation for a quantum particle in a potential $U(z)$.

$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dz^2} + U(z)\psi = E\psi$.

What standard differential equation is derived from the Schrödinger equation for a particle in a uniform force field $U(z) = -Fz$?

The Airy equation: $\frac{d^2\psi}{dx^2} + x\psi = 0$, or $\frac{d^2 f}{dx^2} - xf = 0$.

What recurrence relation is satisfied by the series coefficients of the Airy equation $\psi'' + x\psi = 0$?

$a_2 = 0$ and $a_{n+2} = -\frac{a_{n-1}}{(n+1)(n+2)}$ for $n \ge 1$.

Why do coefficients of the form $a_{3k+2}$ vanish in the series solution of the Airy equation?

Because $a_2 = 0$, and every third successive term in that progression is proportional to $a_2$.

What two standard linearly independent special functions form the general solution to the Airy equation $f'' - xf = 0$?

The Airy functions $\text{Ai}(x)$ and $\text{Bi}(x)$.

How do the Airy functions $\text{Ai}(x)$ and $\text{Bi}(x)$ behave for $x < 0$ and $x > 0$?

For $x < 0$, both oscillate with decaying amplitude; for $x > 0$, $\text{Ai}(x)$ decays exponentially to zero while $\text{Bi}(x)$ grows exponentially without bound.

What two special functions are plotted in this graph, showing oscillatory decay for negative arguments and diverging/decaying exponentials for positive arguments?

The Airy functions $\text{Ai}(x)$ (blue) and $\text{Bi}(x)$ (red).

What potential energy function defines the 1D quantum harmonic oscillator of frequency $\omega$?

$U(z) = \frac{1}{2}m\omega^2 z^2$.

What substitution transforms the nondimensional harmonic oscillator equation $\psi'' - (x^2 - (2N+1))\psi = 0$ into the Hermite equation?

$\psi(x) = f(x)\exp\left(-\frac{1}{2}x^2\right)$.

Write the standard Hermite differential equation for parameter $N$.

$\frac{d^2 f}{dx^2} - 2x\frac{df}{dx} + 2N f = 0$.

What recurrence relation is satisfied by the series coefficients of the Hermite equation?

$a_{n+2} = \frac{2(n-N)}{(n+1)(n+2)}a_n$.

Why does the power series solution to the Hermite equation terminate to form a polynomial when $N$ is an integer?

When $n = N$, the factor $(n - N)$ vanishes, forcing $a_{N+2} = 0$ and causing all subsequent coefficients of that parity to be zero.

What normalisation condition is conventionally applied to the Hermite polynomial $H_N(x)$?

The coefficient of the highest power $x^N$ is set to $2^N$.

What are the first four Hermite polynomials, $H_0(x), H_1(x), H_2(x),$ and $H_3(x)$?

$H_0(x) = 1$, $H_1(x) = 2x$, $H_2(x) = 4x^2 - 2$, and $H_3(x) = 8x^3 - 12x$.

How is the double factorial $n!!$ defined for an odd integer $n = 2p - 1$?

$(2p - 1)!! = \frac{(2p)!}{2^p p!}$.

What is the formula for the normalized stationary wavefunctions $\psi_N(x)$ of the 1D quantum harmonic oscillator?

$\psi_N(x) = \frac{1}{\sqrt{2^N N!\sqrt{\pi}}} H_N(x)\exp\left(-\frac{1}{2}x^2\right)$.

What physical quantum system's energy levels, wavefunctions, and confining potential are shown in this diagram?

The 1D quantum harmonic oscillator stationary states $\psi_N(x)$ in a parabolic potential.

Write the Legendre differential equation for integer $N$.

$(1-x^2)\frac{d^2 f}{dx^2} - 2x\frac{df}{dx} + N(N+1)f = 0$.

What recurrence relation is obtained when substituting a power series into the Legendre equation?

$a_{n+2} = \frac{(n-N)(n+N+1)}{(n+1)(n+2)}a_n$.

What standard normalisation condition is chosen to define the Legendre polynomials $P_N(x)$?

$P_N(1) = 1$.

Write the Laguerre differential equation for integer $N$.

$x\frac{d^2 f}{dx^2} + (1-x)\frac{df}{dx} + Nf = 0$.

What recurrence relation is satisfied by the series coefficients of the Laguerre equation?

$a_{n+1} = \frac{n-N}{(n+1)^2}a_n$.

In what prominent chemical application do Laguerre polynomials arise?

In describing the radial part of the electronic wavefunctions for the hydrogen atom.

How is the Gamma function $\Gamma(x)$ defined for real $x > 0$, and what recurrence relation does it satisfy?

$\Gamma(x) = \int_0^\infty t^{x-1}e^{-t}\,dt$, satisfying $\Gamma(x+1) = x\Gamma(x)$.

Write Bessel's differential equation with parameter $\zeta$.

$x^2\frac{d^2 f}{dx^2} + x\frac{df}{dx} + (x^2 - \zeta^2)f = 0$.

Why does the standard Maclaurin series method fail at $x = 0$ for Bessel's equation, and what method resolves this?

Because $x = 0$ is a regular singular point where the coefficient of $f''$ vanishes; the Frobenius method circumvents this.

What family of special functions is plotted here for orders $0$ through $5$, illustrating oscillatory behavior and infinite sets of zeros?

Bessel functions of the first kind, $J_0(x)$ through $J_5(x)$.