Math Cribsheet

Trigonometry
Fourier
Series
Misc

Addition Formulas:

(1)

$\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta \ \cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta \ \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta \ \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta \ \tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta} \ \tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta} \\$

Law of Cosines:

(2)

$c^2=a^2+b^2-2ab\cos\theta$

Law of Sines:

(3)

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Double Angle Formulas:

(4)

$\sin2\alpha=2\sin\alpha\cos\alpha \ \cos2\alpha=\cos^2\alpha-\sin^2\alpha \ =2\cos^2\alpha-1 \ =1-2\sin^2\alpha$

Half Angle Formulas:

(5)

$\cos^2\frac{\alpha}{2}=\frac{1+\cos\alpha}{2} \ \sin^2\frac{\alpha}{2}=\frac{1-\cos\alpha}{2} \\$

Product Formulas:

(6)

$\sin\alpha\cos\beta=\frac{1}{2}[\sin(\alpha-\beta)+\sin(\alpha+\beta)] \ \sin\alpha\sin\beta=\frac{1}{2}[\cos(\alpha-\beta)-\cos(\alpha+\beta)] \ \cos\alpha\cos\beta=\frac{1}{2}[\cos(\alpha-\beta)+\cos(\alpha+\beta)] \\$

Miscellaneous

(7)

$\sin^2\alpha+\cos^2\alpha=1 \ \cos\left(\frac{\pi}{2}-\alpha\right)=\sin\alpha \ \sin\left(\frac{\pi}{2}-\alpha\right)=\cos\alpha \ \tan^2\theta+1=\sec^2\theta \ 1+\cot^2\theta=\csc^2\theta \\$

Fourier Series

Decomposition of a function as a summation of sines.

(8)

$g_\theta(\theta)=\sum\limits_{k=-\infty}^\infty a_k\mathrm{e}^{ik\theta} \ \mbox{where} \ a_k=\dfrac{1}{2\pi}\int_0^{2\pi}g_\theta(\theta)\mathrm{e}^{-ik\theta}d\theta$

Fourier Transform

(9)

$\mathcal{F}\{f(x,y)\}=F(f_x,f_y)=\iint_{-\infty}^{\infty}f(x,y)\mathrm{e}^{-2\pi i(f_xx+f_yy)}dx\,dy$

Inverse Fourier Transform

(10)

$\mathcal{F}^-1\{F(f_x,f_y)\}=f(x,y)=\iint_{-\infty}^{\infty}F(f_x,f_y)\mathrm{e}^{2\pi i(f_xx+f_yy)}df_x\,df_y$

Fourier-Bessel Transform [3]

A function $g(r,\theta)$ is said to be circularly symmetric if: $g(r,\theta)=g_R(r)$
Transform the Fourier transform into polar coordinates using:

(11)

$r=\sqrt{x^2+y^2} & \theta=\tan^{-1}\dfrac{y}{x} & x=r\cos\theta & y=r\sin\theta \ \rho=\sqrt{f_x^2+f_y^2} & \phi=\tan^{-1}\dfrac{f_y}{f_x} & f_x=\rho\cos\phi & f_y=\rho\sin\phi \\$

The Fourier transform in polar form is called the Fourier-Bessel transform or alternatively the Hankel transform (of zero order).

(12)

$\mathcal{B}\{g_R(r)\}=G_0(\rho,\phi)=\int_0^\infty r\,g_R(r)\:dr\int_0^{2\pi}\mathrm{e}^{-2\pi i r \rho \cos(\theta-\phi)}\:d\theta$

Bessel function

Bessel function of the first kind, zero order

(13)

$J_0(a)={1\over 2\pi}\int_0^{2\pi}\mathrm{e}^{-ia\cos(\theta-\phi)}d\theta$

Apply the bessel function to Eq. (12):

(14)

$\mathcal{B}\{g_R(r)\}=G_0(\rho,\phi)=G_0(\rho)=2\pi\int_0^\infty r\,g_R(r)J_0(2\pi r \rho)\:dr$

And the inverse:

(15)

$\mathcal{B}^{-1}\{G_0(\rho)\}=g_R(r)=2\pi\int_0^\infty\rho G_0(\rho) J_0(2\pi r \rho)\,d\rho$

Si Function

See Ref. [4] pp. 231-232: 5.2.1 and 5.2.25

(16)

$\mathrm{Si}(z)=\int_0^z\frac{\sin x}{x}dx \ \int_0^\infty\frac{\sin x}{x}dx=\frac{\pi}{2} \\$

Error function (Gaussian)

See Ref. [4] pp. 297-298: 7.1.1 and 7.1.16

(17)

$\mathrm{erf}(z)=\frac{2}{\sqrt{\pi}}\int_0^z\mathrm{e}^{-x^2}dx \ \int_{-\infty}^\infty\mathrm{e}^{-x^2}dx = \sqrt{\pi} \ \int_{-\infty}^\infty\mathrm{e}^{-ax^2}dx = \sqrt{\dfrac{\pi}{a}} \ \int_{-\infty}^\infty\mathrm{e}^{-ax^2-bx}dx = \sqrt{\dfrac{\pi}{a}}\mathrm{e}^{b^2/4a} \\$

Delta Function

2D Property

(18)

$\delta(ax,by)={1\over|ab|}\delta(x,y)$

Useful Delta function

(19)

$\delta(f_x)=\int_{-\infty}^\infty\mathrm{e}^{2\pi if_x}dx$

Dirac Delta function

See Ref. [3], Appendix 1 p. 393.

(20)

$\int_{-\infty}^\infty\delta(\xi-b)h(\xi)d\xi=\left\{\begin{array}{ll} h(b) & \mbox{$b$ a point of continuity of $h$} \ {1\over 2}\left[h(b^+)+h(b^-)\right] & \mbox{$b$ a point of discontinuity of $h$} \end{array}$

Euler

(21)

$\mathrm{e}^{iy}=\cos y + i\sin y \ \cos(y)=\frac{1}{2}\left(\mathrm{e}^{iy}+\mathrm{e}^{-iy}\right) \ \sin(y)=\frac{-i}{2}\left(\mathrm{e}^{iy}+\mathrm{e}^{-iy}\right) \\$

Useful functions

See Ref. [3] p. 13.

(22)

$\mathrm{rect}(x) &=& \left\{ \begin{array}{ll} 1 & |x|<\frac{1}{2} \ \frac{1}{2} & |x|=\frac{1}{2} \ 0 & \mbox{otherwise} \end{array} \ \mathrm{sinc}(x) &=& \dfrac{\sin(\pi x)}{\pi x} \ \mathrm{sgn}(x) &=& \left\{ \begin{array}{ll} 1 & x>0 \ 0 & x=0 \ -1 & x<0 \end{array} \ \Lambda(x) &=& \left\{ \begin{array}{ll} 1-|x| & |x|\leq 1 \ 0 & \mbox{otherwise} \end{array} \ \mathrm{comb}(x) &=& \sum\limits_{n=-\infty}^\infty\delta(x-n) \ \mathrm{circ}(\sqrt{x^2+y^2}) &=& \left\{ \begin{array}{ll} 1 & \sqrt{x^2+y^2}<1 \ 0 & \sqrt{x^2+y^2}=1 \ 0 & \mbox{otherwise} \end{array} \\$

Fourier Transform Pairs

See Ref. [3] p. 14.

(23)

$\mathrm{rect}(ax)\mathrm{rect}(by) &\stackrel{\mathcal{F}}{\rightarrow}& \dfrac{1}{|ab|}\mathrm{sinc}(f_x/a)\mathrm{sinc}(f_y/b) \ \Lambda(ax)\Lambda(by) &\stackrel{\mathcal{F}}{\rightarrow}& \dfrac{1}{|ab|}\mathrm{sinc}^2(f_x/a)\mathrm{sinc}^2(f_y/b) \ \mathrm{comb}(ax)\mathrm{comb}(by) &\stackrel{\mathcal{F}}{\rightarrow}& \dfrac{1}{|ab|}\mathrm{comb}(f_x/a)\mathrm{comb}(f_y/b) \ \delta(ax,by) &\stackrel{\mathcal{F}}{\rightarrow}& \dfrac{1}{|ab|} \ \mathrm{e}^{\pi i(ax+by)} &\stackrel{\mathcal{F}}{\rightarrow}&\delta(f_x-a/2,f_y-b/2) \ \mathrm{e}^{-\pi (a^2x^2+b^2y^2)} &\stackrel{\mathcal{F}}{\rightarrow}& \dfrac{1}{|ab|}\mathrm{e}^{-\pi \left(\dfrac{f_x^2}{a^2}+\dfrac{f_y^2}{b^2}\right)} \ \mathrm{e}^{\pi i(a^2x^2+b^2y^2)} &\stackrel{\mathcal{F}}{\rightarrow}& \dfrac{i}{|ab|}\mathrm{e}^{-\pi i\left(\dfrac{f_x^2}{a^2}+\dfrac{f_y^2}{b^2}\right)} \ \cos\left(\dfrac{2\pi x}{d}\right)=\tfrac{1}{2}\left[\mathrm{e}^{\frac{2\pi ix}{d}}+\mathrm{e}^{-\frac{2\pi ix}{d}}\right] &\stackrel{\mathcal{F}}{\rightarrow}& \tfrac{1}{2}\delta\left(f_x-\tfrac{1}{d}\right)+\tfrac{1}{2}\delta\left(f_x+\tfrac{1}{d}\right) \ 1+\cos\left(\dfrac{2\pi x}{d}\right) &\stackrel{\mathcal{F}}{\rightarrow}& \delta(f_x) + \mathcal{F}\left\{\ \cos\left(\dfrac{2\pi x}{d}\right) \right\} \\$

Finding a function's spectrum

It is most useful to find the transform of a periodic function by first making sure the function is symmetric about the origin (this will send the fourier sine coeffients to zero later). Then decompose the function into its Fourier series, see Eq. (8. Fourier transforming this decomposed function should just be the spectrum of a summation of cosine components (which yields a fixed frequency comb) and an amplitude envelope function (which modulates the comb). Refer to Ch. 2.5-2.6 in Ref. [6].

Geometric series

A geometric series:

(24)

$a+ar+ar^2+\dotsb+ar^{k-1}+\dotsb\quad(a\neq 0)$

converges if $|r|<1$ and diverges if $|r| \geq 1$ . If the series converges then:

(25)

$\frac{a}{1-r}=a+ar+ar^2+\dotsb+ar^{k-1}+\dotsb$

Taylor series

Taylor series for $f$ about $f=a$ :

(26)

$\sum_{k=0}^\infty {f^{(k)}(a) \over k!} (x-a)^k = f(a) + f'(a)(x-a) + {f''(a)\over 2!}(x-a)^2+\dotsb+{f^{(k)}(a)\over k!}(x-a)^k+\dotsb$

Maclaurin series

The Maclaurin series is just the Taylor series about $f=0$ :

(27)

$\sum_{k=0}^\infty {f^{(k)}(0) \over k!}x^k = f(0) + f'(0)x + {f''(0)\over 2!}x^2+\dotsb+{f^{(k)}(0)\over k!}x^k+\dotsb$

Binomial series

Maclaurin series for $(1+x)^m$ for $|x|<1$ :

(28)

$(1+x)^m = 1+\sum_{k=1}^\infty{m(m-1)\dotsb(m-k+1)\over k!}x^k = 1 + mx + {m(m-1)\over 2!}x^2+\dotsb+{m(m-1)(m-2)\dotsb(m-k+1)\over k!}x^k+\dotsb$

Useful for paraxial approximation:

(29)

$\sqrt{1+x}=1+{1\over 2}x-{1\over 4}\cdot{1\over 2!}x^2+{3\over 8}\cdot{1\over 3!}x^3+\dotsb$

Paraxial conditions: $|x|\ll 1$ then,

(30)

$\sqrt{1+x}\approx 1+\frac{x}{2}$

Expansions

(31)

$\frac{1}{1-x} &= 1+x+x^2+x^3+x^4+\dotsb &= \sum_{k=0}^\infty x^k \ \frac{1}{1-cx} &= 1+cx+c^2x^2+c^3x^3+\dotsb &= \sum_{k=0}^\infty c^k x^k \ \frac{1}{1-x^n} &= 1+x^n+x^{2n}+x^{3n}+\dotsb &= \sum_{k=0}^\infty x^{nk} \ \frac{x}{(1-x)^2} &= x+2x^2+3x^3+4x^4+\dotsb &= \sum_{k=0}^\infty kx^k \ \mathrm{e}^x &= 1+x+{1\over 2!}x^2+{1\over 3!}x^3+\dotsb &= \sum_{k=0}^\infty \frac{x^k}{k!}\ \sin x &= x-{1\over 3!}x^3+{1\over 5!}x^5-{1\over 7!}x^7+\dotsb &= \sum_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k+1)!}\ \cos x &= 1-{1\over 2!}x^2+{1\over 4!}x^4-{1\over 6!}x^6+\dotsb &= \sum_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!}\\$

Vectors

Nomenclature

(32)

$\vec{a}=a_1\hat{x}+a_2\hat{y}+a_3\hat{z}$

Dot Product (Inner Product)

(33)

$\vec{a}\cdot\vec{b}=a_1b_1+a_2b_2+a_3b_3$

(34)

$\|\vec{a}\|=\sqrt{\vec{a}\cdot\vec{a}}$

If the angle between two vectors $\vec{a}$ and $\vec{b}$ is $\theta$ then

(35)

$\vec{a}\cdot\vec{b}=\|\vec{a}\|\|\vec{b}\|\cos\theta$

Cross Product

(36)

$\vec{a}\times\vec{b}=\left| \begin{array}{ccc} \hat{x} & \hat{y} & \hat{z} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ \end{array} \right|$

(37)

$=\left|\begin{array}{cc}a_2 & a_3\\ b_2 & b_3\end{array} \right|\hat{x} -\left|\begin{array}{cc}a_1 & a_3\\ b_1 & b_3\end{array} \right|\hat{y} +\left|\begin{array}{cc}a_1 & a_2\\ b_1 & b_2\end{array} \right|\hat{z} \\$

(38)

$=(a_2b_3-b_2a_3)\hat{x}-(a_1b_3-b_1a_3)\hat{y}+(a_1b_2+b_1a_2)\hat{z}$

Euler

(39)

$\mathrm{e}^{iy}=\cos y + i\sin y \ \cos(y)=\frac{1}{2}\left(\mathrm{e}^{iy}+\mathrm{e}^{-iy}\right) \ \sin(y)=\frac{-i}{2}\left(\mathrm{e}^{iy}+\mathrm{e}^{-iy}\right) \\$

Exponential

(40)

$a^k\cdot b^k=(ab)^k \ \dfrac{a^k}{b^k}=\left(\dfrac{a}{b}\right)^k \ \left(a^k\right)^l=a^{kl} \ a^x=\mathrm{e}^x\ln a\\$

Gaussian

(41)

$I(y)=I_0\mathrm{e}^{-y^2/a^2}$

where $I_0$ is the irradiance maximum. The standard deviation $(\sigma)$ of the
Gaussian is equal to $a/\sqrt{2}$ . Three times the standard deviation
$(\pm3\sigma)$ is $\pm3a/\sqrt{2}$ and contains 99.73% of the area.