MatheMaticas中的巧妙范例[转载自哆嗒数学平台吧chzhn]

BlockRandom[SeedRandom[11]; 
Count[Table[walkerPosition = {0, 0, 0}; steps = 0; 
While[steps == 0 || (steps < 100 && walkerPosition =!= {0, 0, 0}), 
steps++; 
walkerPosition = 
walkerPosition + {{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {0, -1, 
0}, {0, 0, 1}, {0, 0, -1}}[[Random[Integer, {1, 6}]]]]; 
steps, {1000}], _?(# < 100 &)]]

\[\left| {\begin{array}{*{20}{c}}{\frac{1}{{B\left( {1,1} \right)}}}&{\frac{1}{{B\left( {1,2} \right)}}}& \cdots &{\frac{1}{{B\left( {1,n} \right)}}}\\{\frac{1}{{B\left( {2,1} \right)}}}&{\frac{1}{{B\left( {2,2} \right)}}}& \cdots &{\frac{1}{{B\left( {2,n}\right)}}}\\{\frac{1}{{B\left( {3,1} \right)}}}&{\frac{1}{{B\left( {3,2} \right)}}}& \cdots &{\frac{1}{{B\left( {3,n} \right)}}}\\\cdots & \cdots & \cdots & \cdots \\{\frac{1}{{B\left( {n,1} \right)}}}&{\frac{1}{{B\left( {n,2} \right)}}}& \cdots &{\frac{1}{{B\left( {n,n} \right)}}}\end{array}} \right| = n!.\]

3.Binomial

希尔伯特矩阵的逆：\[{H_{ij}} = \frac{1}{{i + j - 1}}.\]

逆矩阵系数为

\[{\left( {{H^{ - 1}}} \right)_{ij}} = {\left( { - 1} \right)^{i + j}}\left( {i + j - 1} \right)\left( \begin{array}{l}n + i - 1\\n - j\end{array} \right)\left( \begin{array}{l}n + j - 1\\n - i\end{array} \right){\left( \begin{array}{l}i + j - 2\\i - 1\end{array} \right)^2}.\]

4.Erf

\[\frac{1}{{1 + \frac{1}{{1 + \frac{2}{{1 + \frac{3}{{1 + \frac{4}{{1 + \frac{5}{{1 + \frac{6}{{1 + \frac{7}{{1 + \frac{8}{{1 + \frac{9}{{1 +  \cdots }}}}}}}}}}}}}}}}}}}} = \sqrt {\frac{{\pi e}}{2}} \left( {1 - \rm{Erf}\left( {\frac{1}{{\sqrt 2 }}} \right)} \right).\]

5.HermiteH

广义 Lissajous 图形：

Block[{n = 11, m = 13},

ParametricPlot[{ Exp[-x^2/2] HermiteH[n, x]/Sqrt[2^n n!],

Exp[-x^2/2] HermiteH[m, x]/Sqrt[2^m m!]}, {x, -8, 8}]]

 

6.BesselI

等差数列的连分数表示

\[1 + \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{4 + \frac{1}{{5 + \frac{1}{{6 +  \cdots }}}}}}}}}} = \frac{{{I_0}\left( 2 \right)}}{{{I_1}\left( 2 \right)}}.\]

代码ContinuedFraction[BesselI[0, 2]/BesselI[1, 2], 20]

输出

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

7.Floor

数列$\{1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6\cdots\}的通项

\[{a_n} = \left[ {\sqrt {2k}  + \frac{1}{2}} \right].\]

8.Exp

（1）Exp迭代分形

DensityPlot[

Length @FixedPointList[

If[TrueQ[Abs[#] > 10.^5], Indeterminate, Exp[#/(x + I y)]] &,

x + I y, 10], {x, -1, 3}, {y, -1, 1}, MaxRecursion -> 4]

（2）黎曼- 维尔斯特拉斯函数任何位置不可微：

ParametricPlot[{Re[#], Im[#]} &@\!\(

\*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(200\)]

\*FractionBox[

SuperscriptBox[\(E\), \(I\

\*SuperscriptBox[\(j\), \(3\)]\ \[CurlyPhi]\)],

SuperscriptBox[\(j\), \(2\)]]\), {\[CurlyPhi], 0, 2 \[Pi]}]

9.Tan

\[1 + \frac{1}{{1 + \frac{1}{{1 + \frac{1}{{3 + \frac{1}{{1 + \frac{1}{{5 + \frac{1}{{1 + \frac{1}{{7 +  \cdots }}}}}}}}}}}}}} = \tan 1.\]

10.Tanh

\[\frac{1}{{1 + \frac{1}{{3 + \frac{1}{{5 + \frac{1}{{7 + \frac{1}{{9 + \frac{1}{{11 + \frac{1}{{13 +  \cdots }}}}}}}}}}}}}} = \tanh 1.\]

11.sinc

\[\int_0^\infty  {\left( {\prod\limits_{k = 0}^n {\rm{sinc}\left( {\frac{x}{{2k + 1}}} \right)} } \right)dx}  = \pi \left( {n = 0,1,2,3,4,5,6} \right)\]

12.Factor

$x^n-1$在整数范围内充分分解后，几乎所有因式的系数不是1就是-1，但是也有一些例外，第一个例子是

\begin{align*}&{x^{105}} - 1 = ( - 1 + x)(1 + x + {x^2})(1 + x + {x^2} + {x^3} + {x^4})(1 + x + {x^2} + {x^3} + {x^4} + {x^5} + {x^6})\\&(1 - x + {x^3} - {x^4} + {x^5} - {x^7} + {x^8})(1 - x + {x^3} - {x^4} + {x^6} - {x^8} + {x^9} - x^{11} + x^{12})\\&(1 - x + {x^5} - {x^6} + {x^7} - {x^8} + x^{10} - {x^{11}} + {x^{12}} - {x^{13}} + {x^{14}} - {x^{16}} + {x^{17}} - {x^{18}} + {x^{19}} - {x^{23}} + {x^{24}})\\&(1 + x + {x^2} - {x^5} - {x^6} - 2{x^7} - {x^8} - {x^9} + {x^{12}} + {x^{13}} + {x^{14}} + {x^{15}} + {x^{16}} + {x^{17}} - {x^{20}} - {x^{22}} - {x^{24}} \\&- {x^{26}} - {x^{28}} + {x^{31}} + {x^{32}} + {x^{33}} + {x^{34}} + {x^{35}} + {x^{36}} - {x^{39}} - {x^{40}} - 2{x^{41}} - {x^{42}} - {x^{43}} + {x^{46}} + {x^{47}} + {x^{48}}).\end{align*}

参考来源：http://tieba.baidu.com/p/2978798364