REVISION
Formula sheet
Everything from the five chapters on one page. Press Ctrl+P for a print-friendly version.
1 · Laplace
| f(t) | F(s) |
|---|---|
| \(1,\ t^n\) | \(\frac1s,\ \frac{n!}{s^{n+1}}\) |
| \(e^{at}\) | \(\frac1{s-a}\) |
| \(\sin at,\ \cos at\) | \(\frac a{s^2+a^2},\ \frac s{s^2+a^2}\) |
| \(\sinh at,\ \cosh at\) | \(\frac a{s^2-a^2},\ \frac s{s^2-a^2}\) |
| \(e^{at}f(t)\) | \(F(s-a)\) |
| \(u(t-a)f(t-a)\) | \(e^{-as}F(s)\) |
| \(\delta(t-a)\) | \(e^{-as}\) |
| \(f(\alpha t)\) | \(\frac1\alpha F(s/\alpha)\) |
| \(y',\ y''\) | \(sY-y_0,\ s^2Y-sy_0-y_0'\) |
| \(\int_0^tf\) | \(F/s\) |
| \(t^nf\) | \((-1)^nF^{(n)}(s)\) |
| \(f/t\) | \(\int_s^\infty F\) |
| periodic, T | \(\frac{\int_0^Te^{-st}f}{1-e^{-sT}}\) |
| \(f*g=\int_0^tf(\tau)g(t-\tau)d\tau\) | \(FG\) |
Transfer function \(H=Y/X\) (zero ICs). Impulse response \(h=\mathcal L^{-1}\{H\}\).
2 · Fourier series (period 2ℓ)
\[f\sim\tfrac{a_0}2+\sum a_n\cos\tfrac{n\pi x}\ell+b_n\sin\tfrac{n\pi x}\ell\] \[a_n=\tfrac1\ell\!\int_{-\ell}^{\ell}\!f\cos\tfrac{n\pi x}\ell,\quad b_n=\tfrac1\ell\!\int_{-\ell}^{\ell}\!f\sin\tfrac{n\pi x}\ell\]Even: \(b_n=0\), \(a_n=\frac2\ell\int_0^\ell\). Odd: \(a_n=0\), \(b_n=\frac2\ell\int_0^\ell\). Half-range cosine is the even extension and half-range sine is the odd extension.
At a jump the series gives \(\frac{f(x^-)+f(x^+)}2\).
\[\tfrac1\ell\int_{-\ell}^{\ell}f^2=\tfrac{a_0^2}2+\sum(a_n^2+b_n^2)\]
\[c_n=\tfrac1{2\ell}\int_{-\ell}^{\ell}fe^{-in\pi x/\ell},\quad c_n=\tfrac{a_n-ib_n}2\]
Useful integrals: \(\int_0^\pi x\sin nx=\frac{\pi(-1)^{n+1}}n\), \(\int_0^\pi x\cos nx=\frac{(-1)^n-1}{n^2}\), \(\int_0^\pi x^2\cos nx=\frac{2\pi(-1)^n}{n^2}\).
Known sums: \(\sum\frac1{n^2}=\frac{\pi^2}6\), \(\sum_{odd}\frac1{n^2}=\frac{\pi^2}8\), \(\sum\frac1{n^4}=\frac{\pi^4}{90}\).
Known sums: \(\sum\frac1{n^2}=\frac{\pi^2}6\), \(\sum_{odd}\frac1{n^2}=\frac{\pi^2}8\), \(\sum\frac1{n^4}=\frac{\pi^4}{90}\).
3 · PDEs
\(B^2-4AC\): \(<0\) elliptic · \(=0\) parabolic · \(>0\) hyperbolic.
| equation | separated solution (ends 0) |
|---|---|
| Heat \(u_t=c^2u_{xx}\) | \(\sum B_n\sin\frac{n\pi x}Le^{-c^2n^2\pi^2t/L^2}\) |
| Wave \(y_{tt}=c^2y_{xx}\) | \(\sum\sin\frac{n\pi x}L(C_n\cos+D_n\sin)\frac{n\pi ct}L\) |
| Laplace, rectangle | \(\sum A_n\sin\frac{n\pi x}a\sinh\frac{n\pi y}a\) |
| Laplace, semi-infinite | \(\sum A_n\sin\frac{n\pi x}ae^{-n\pi y/a}\) |
| Laplace, polar | \(\sum r^n(a_n\cos n\theta+b_n\sin n\theta)\) |
Coefficients: \(B_n=\frac2L\int_0^Lf\sin\frac{n\pi x}L\). For initial velocity \(g\): \(D_n\frac{n\pi c}L=\frac2L\int_0^Lg\sin\frac{n\pi x}{L}\). Sine series of a constant \(K\): \(\frac{2K(1-(-1)^n)}{n\pi}\).
Non-zero ends: subtract the steady line \(u_s\) first, then solve for \(u-u_s\).
4 · Series solutions
Ordinary: \(P_0(a)\ne0\). Regular singular: \(P_0(a)=0\) with \((x-a)\frac{P_1}{P_0}\) and \((x-a)^2\frac{P_2}{P_0}\) analytic.
\[y'=\sum(n+1)a_{n+1}x^n,\quad y''=\sum(n+2)(n+1)a_{n+2}x^n\]
Frobenius \(y=\sum a_nx^{n+m}\). Lowest power gives the indicial equation \(m(m-1)+p_0m+q_0=0\).
Case 1 non-integer gap: two series. Case 2 equal roots: \(y_1\ln x+\dots\). Case 3 integer gap: \(Cy_1\ln x+\dots\), and C may be 0.
\[J_n=\sum_r\frac{(-1)^r}{r!(n+r)!}\left(\frac x2\right)^{2r+n}\]
\[P_n=\frac{1}{2^nn!}\frac{d^n}{dx^n}(x^2-1)^n,\quad\int_{-1}^1P_mP_n=0\]
Case 1 non-integer gap: two series. Case 2 equal roots: \(y_1\ln x+\dots\). Case 3 integer gap: \(Cy_1\ln x+\dots\), and C may be 0.
\(P_2=\frac12(3x^2-1)\), \(P_3=\frac12(5x^3-3x)\), \(P_4=\frac18(35x^4-30x^2+3)\), \(P_5=\frac18(63x^5-70x^3+15x)\).
5 · Fourier transform
\[F=\int_{-\infty}^\infty fe^{-i\omega t}dt,\qquad f=\frac1{2\pi}\int_{-\infty}^\infty Fe^{i\omega t}d\omega\]| f(t) | F(ω) |
|---|---|
| \(u(t)e^{-\alpha t}\) | \(\frac1{\alpha+i\omega}\) |
| box \(|t|\le\alpha\) | \(\frac{2\sin\alpha\omega}{\omega}\) |
| \(e^{-\alpha|t|}\) | \(\frac{2\alpha}{\alpha^2+\omega^2}\) |
| \(e^{-\alpha t^2}\) | \(\sqrt{\pi/\alpha}\,e^{-\omega^2/4\alpha}\) |
| \(\delta(t-a),\ 1\) | \(e^{-i\omega a},\ 2\pi\delta(\omega)\) |
| \(\cos at\) | \(\pi[\delta(\omega-a)+\delta(\omega+a)]\) |
| \(\operatorname{sgn}t\) | \(\frac{2}{i\omega}\) |
| property | result |
|---|---|
| \(f(t-t_0)\) | \(e^{-i\omega t_0}F\) |
| \(e^{iat}f\) | \(F(\omega-a)\) |
| \(f(\alpha t)\) | \(\frac1\alpha F(\omega/\alpha)\) |
| \(f'\), \(tf\) | \(i\omega F\), \(iF'\) |
| \(f*g\) | \(FG\) |
| \(F(t)\) | \(2\pi f(-\omega)\) |
\(F_c=\int_0^\infty f\cos\omega t\) and \(F_s=\int_0^\infty f\sin\omega t\), with inverses \(\frac2\pi\int_0^\infty\). In a PDE, \(\mathcal F_x\{u_{xx}\}=-\omega^2\hat u\) and \(\mathcal F_s\{u_{xx}\}=-\omega^2U_s\) (when \(u(0)=0\)).