Highly Useful Formulas

The following formulas can make problem solving much more convenient.

I. Gaussian Integrals

\[I_n = \int_{0}^{\infty}x^n e^{-b x^2}\mathrm{d}x %= %\left\{\begin{aligned} %\end{aligned}\right. \]

Here is a convenient way to remember it:

First, remember

\[I_0 = \frac{1}{2}\sqrt{\frac{\pi}{b}},\quad I_1 = \frac{1}{2}\frac{1}{b}\]

Next,

\[I_{n+2}=\frac{n+1}{2b}I_n\]

This formula applies to both odd and even cases of \(n\).

If you are dealing with Gaussian functions (for example, in probability theory), then this integral will appear very frequently!

II. Powers of Trigonometric Functions

\[I_n=\int_{0}^{\frac{\pi}{2}}\sin^{n}x\mathrm{d}x = \int_{0}^{\frac{\pi}{2}}\cos^{n}x\mathrm{d}x\]

The following expression gives \(I_n\)

\[I_n = \left\{\begin{aligned} &\frac{(n-1)!!}{n!!}&, \quad n \text{ is odd} \\ &\frac{(n-1)!!}{n!!}\frac{\pi}{2}&, \quad n \text{ is even} \end{aligned}\right.\]

This formula is very useful when integrating \(\theta,\,\varphi\) in spherical coordinates!

III. The Laplace Transform of x^n

\[I_n=\int_{0}^{\infty}x^n\mathrm{e}^{-kx}\mathrm{d}x\]

The following expression gives \(I_n\):

\[I_n=\frac{n!}{k^{n+1}}\]

This formula is useful when integrating \(r\) in spherical coordinates (if you are dealing with terms containing \(\mathrm{e}^{kr}\))

IV. A Shortcut for Integration by Parts

\[I=\int_{a}^{b} u(x)v(x)\mathrm{d}x\]

The diagram for the shortcut is as follows:

\[\begin{alignedat}{6} & u(x) & \quad & u^{(1)}(x) & \quad & u^{(2)}(x) & \quad & u^{(3)}(x) & \quad & \cdots \\ & & \searrow^{+} & & \searrow^{-} & & \searrow^{+} & & \searrow^{-} \\ & v(x) & \quad & v^{(-1)}(x) & \quad & v^{(-2)}(x) & \quad & v^{(-3)}(x) & \quad & \cdots & & \end{alignedat}\]

Here, \(u^{(n)}(x)\) denotes the \(n\)th derivative of \(u(x)\), and \(v^{(-n)}(x)\) denotes the \(n\)th indefinite integral of \(v(x)\).

In other words,

\[\begin{aligned} I&=\int_{a}^{b} u(x)v(x)\mathrm{d}x \\&= \left.u(x)v^{(-1)}(x)\right|^{b}_{a} - \left.u^{(1)}(x)v^{(-2)}(x)\right|^{b}_{a}+\left.u^{(2)}(x)v^{(-3)}(x)\right|_{a}^{b} - \cdots \end{aligned}\]

This process continues until \(u^{(n)}(x)=0\).

Or, you may also stop manually at a certain point, for example
\(\begin{aligned} I&=\int_{a}^{b} u(x)v(x)\mathrm{d}x \\&= \left.u(x)v^{(-1)}(x)\right|^{b}_{a} - \left.u^{(1)}(x)v^{(-2)}(x)\right|^{b}_{a}+\int_{a}^{b}u^{(2)}(x)v^{(-2)}(x)\mathrm{d}x \end{aligned}\)
In this case, you want \(\int_{a}^{b}u^{(2)}(x)v^{(-2)}(x)\mathrm{d}x\) to contain \(I\), so that you can rearrange the terms to solve for \(I\)

V. Differential Operators in Spherical Coordinates

In orthogonal curvilinear coordinates, we have

\[\nabla = \frac{\bm{e}_{1}}{h_1}\frac{\partial}{\partial q_1}+\frac{\bm{e}_{2}}{h_2}\frac{\partial}{\partial q_2}+\frac{\bm{e}_{3}}{h_3}\frac{\partial}{\partial q_3}\]\[\nabla \cdot \mathbf{A} = \frac{1}{h_1h_2h_3}\left[\frac{\partial}{\partial q_1}(A_1h_2h_3)+\frac{\partial}{\partial q_2}(h_1A_2h_3)+\frac{\partial}{\partial q_3}(h_1h_2A_3)\right]\]\[\nabla\times\mathbf{A}=\frac{1}{h_1h_2h_3}\begin{vmatrix} h_1\bm{e_1} & h_2\bm{e_2} & h_3\bm{e_3} \\ \frac{\partial}{\partial q_1} & \frac{\partial}{\partial q_2} & \frac{\partial}{\partial q_3} \\ h_1A_1 & h_2A_2 & h_3A_3 \end{vmatrix}\]\[\begin{aligned} \nabla^2u&=\nabla\cdot(\nabla u)\\&= \frac{1}{h_1h_2h_3}\left[\frac{\partial}{\partial q_1}\left(\frac{h_2h_3}{h_1}\frac{\partial}{\partial q_1}\right) + \frac{\partial}{\partial q_2}\left(\frac{h_1h_3}{h_2}\frac{\partial}{\partial q_2}\right) + \frac{\partial}{\partial q_3}\left(\frac{h_1h_2}{h_3}\frac{\partial }{\partial q_3}\right) \right]u \end{aligned}\]

Substituting

\[\begin{matrix} h_1 \rightarrow 1 & h_2 \rightarrow r & h_3 \rightarrow r\sin\theta \\ \bm{e}_1 \rightarrow \bm{\widehat{r}} & \bm{e}_2 \rightarrow \bm{\widehat{\theta}} & \bm{e}_3 \rightarrow \bm{\widehat{\varphi}} \end{matrix}\]

gives the formulas in spherical coordinates.

In fact, \(h_1,\,h_2,\,h_3\) are called the Lamé coefficients.

With this trick, you can derive PDEs in spherical coordinates from scratch during an exam.

VI. The Exponential of a Matrix

Actually, this is easy to derive… it is included here just to round out the list:

\[\exp\left(t\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}\right) =\begin{bmatrix} \cos t & -\sin t \\ \sin t & \cos t \end{bmatrix}\]\[\exp\left(t\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\right) =\begin{bmatrix} \cosh t & \sinh t \\ \sinh t & \cosh t \end{bmatrix}\]