本文主要记录一下在使用 Mathematica 过程中常用的 build-in functions.
Numeric
| Log | Log[$z$] gives the natural logarithm of $z$ (logarithm to base $e$). Log[$b,z$] gives the logarithm to base $b$. — Log[E^2] -> 2Log[10.] -> 2.30259Log[2, 4] -> 2 |
| N | N[$expr$] gives the numerical value of $expr$. N[$expr, n$] attempts to give a result with $n$‐digit precision. — N[1/7] -> 0.142857N[Gamma[3.3], 50] -> 2.68344(Machine presicion input will lead to only machine precision output) N[Gamma[33/10], 50] -> 2.6834373819557687935963273147667112586281870043548 |
Calculus
| D | D[$f,x$] gives the partial derivative $\partial f/ \partial x$. D[$f, \{x, n\}$] gives the multiple derivative $\partial^n f/ \partial x^n$. D[$f, x, y, \dots$] gives the partial derivative $\cdots (\partial / \partial y) (\partial / \partial x) f$. D[$f, \{x, n\}, \{y, m\}, \dots$] gives the multiple partial derivative $\cdots (\partial^m / \partial y^m) (\partial^n / \partial x^n) f$. D[$f, \{\{x_1, x_2, \dots\}\}$] for a scalar $f$ gives the vector derivative $(\partial f/ \partial x_1, \partial f/ \partial x_2, \cdots)$. |
| DSolve | |
Series
| GeneratingFunction | GeneratingFunction[$expr, n, x$] gives the generating function in $x$ for the sequence whose $n$-th series coefficient is given by the expression $expr$. GeneratingFunction[$expr, \{n_1, n_2, \dots\}, \{x_1, x_2, \dots\}$] gives the multidimensional generating function in $x_1, x_2, \dots$ whose $n_1, n_2, \dots$ coefficient is given by $expr$. — GeneratingFunction[$\frac{1}{(n+1)!m!}, \{n, m\}, \{x, y\}$] -> $\frac{e^y(-1+e^x)}{x}$ |
| Series | Series[$f, \{x, x_0, n\}$] generates a power series expansion for $f$ about the point $x=x_0$ to order $(x-x_0)^n$, where $n$ is an explicit integer. Series[$f, x\to x_0$] generates the leading term of a power series expansion for $f$ about the point $x=x_0$. Series[$f, \{x, x_0, n_x\}, \{y, y_0, n_y\}, \dots$] successively finds series expansions with respect to $x$, then $y$, etc. |
| SeriesCoefficient | SeriesCoefficient[$series, n$] finds the coefficient of the $n$-th-order term in a power series in the form generated by Series. SeriesCoefficient[$f, \{x, x_0, n\}$] finds the coefficient of $(x-x_0)^n$ in the expansion of $f$ about the point $x=x_0$. SeriesCoefficient[$f, \{x, x_0, n_x\}, \{y, y_0, n_y\}, \dots$] finds a coefficient in a multivariate series. |
例子
找 OBGF 的一个系数：
$f(z,u)$ 是关于计数 不出现00的长度为$n$的01序列中 0 的出现次数 的 OBGF，下面计算的是 0 的期望次数趋近 $\frac{n}{\sqrt{5}\phi}$，其中 $\phi=\frac{1+\sqrt{5}}{2}$。
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