Mathematica函數速查手冊

事先聲明，本手冊既不完整，又不準確。本手冊是從Mathematica幫助文檔中的MATHEMATICS AND ALGORITHMS和VISUALIZATION AND GRAPHICS中節選，再加上了我的一些翻譯而完成。

Shift+Enter用於輸入表達式

Crtl+K用於打開輸入提示

用於展開代數式

Expand[poly,patt]            用變量patt展開多項式poly

ExpandAll[expr]            徹底展開表達式

對代數式因式分解

N用於將符號轉爲數值

              將表達式expr的x用value替換

    perform several replacements

                     定義變量x並賦值value

                          將變量x清除

Simplify[expr]用於化簡表達式

Simplify[expr,assum]        simplify  expr  with assumptions

FullSimplify用於更大範圍的化簡

Together[expr]             通分

Apart[expr]                部分分式展開

Apart[expr,var]       以變量var對錶達式expr進行部分分式展開

Cancel[expr]               分式約分

Collect[expr,x]             將表達式按x合併同類項

Collect[poly,{x,y,…}]      arrange a polynomial as a sum of powers of x, y, …

Collect[poly,patt,h]         apply h to each final coefficientobtained

FactorTerms[expr,x]         pull out factors that do not depend on  x

FactorTerms[poly]  提取公因式

FactorTerms[poly,{x,y,…}] pull out any overall factorthat does not depend on x, y, …

TrigExpand[expr]           三角函數表達式的展開

TrigFactor[expr]           三角函數表達式的因式分解

TrigReduce[expr]           三角函數化簡

TrigToExp[expr]            將三角函數轉化爲指數形式

ExpToTrig[expr]            將指數形式轉換爲三角函數形式

FunctionExpand[expr]       expand out special and other functions

ComplexExpand[expr]        將表達式按複數展開

PowerExpand[expr]          將(x y)^p 轉換爲x^p y^p

PowerExpand[expr,Assumptions->assum]   expand out  expr assuming  assum

Element[x,dom]             聲明x屬於dom域

Element[{x₁,x₂,…},dom]     state that all the  x_i are elements of the domain  dom

Reals                     實數

Integers                  整數

Primes                    素數

Coefficient[expr,form]      求表達式中變量的係數

Exponent[expr,form] 求表達式expr中變量form的最大指數

Part[expr,n]或expr[[n]] 求表達式expr的第n項

Numerator[expr] 求表達式expr的分子

Denominator[expr] 求表達式expr的分母

Postfix[f[expr]]
prints with  f[expr] given in default postfixform:  expr//f

                   執行command並且不輸出

expr//Short                顯示錶達式輸出的一行

Short[expr,n]              顯示錶達式輸出的n行

     讀入文件包

HornerForm[expr,x]  使表達式expr以x的嵌套形式展現

PolynomialQ[expr,x] 查詢expr是否是x的多項式

PolynomialQ[expr,{x₁,x₂,…}]  test whether  expr is a polynomial in the  x_i

Variables[poly]  取得多項式poly的變量

Coefficient[poly,expr]  求多項式poly中表達式expr的係數

MonomialList[poly]     獲取多項式poly的單項式

CoefficientRules[poly]  獲取多項式poly的單項式的指數向量和係數

ExpandNumerator[expr]  只展開分子

ExpandDenominator[expr] 只展開分母

PolynomialQuotient[ ,q(x),x]    對於  ，求得

PolynomialRemainder[ ,q(x),x]    對於  ，求得b(x)

PolynomialGCD[poly₁,poly₂]  求最大公約數

PolynomialMod[poly,m]      化簡poly，通過用m減去某多項式的m倍

Factor[poly,Extension->{a₁,a₂,…}]  因式分解是允許引入a₁,a₂,…

Factor[poly,Extension->Automatic]  因式分解允許引入代數數

LogicalExpand[expr]     將邏輯函數展開

PiecewiseExpand[expr]  分段函數展開

PiecewiseExpand[expr,assum]  expand out with the specifiedassumptions

    賦值

   判斷x和y是否相等

Solve[lhs==rhs,x]  解方程

           use the list of rules to get values for  x

        usethe list of rules to get values for an expression

Solve[{lhs₁==rhs₁,lhs₂==rhs₂,…},{x,y,…}]   解方程組

Solve[eqns,vars,elims]     find solutions for  vars, eliminating the variables  elims

SolveAlways[eqns,vars]  求使方程恆成立的係數值

Eliminate[{lhs₁==rhs₁,lhs₂==rhs₂,…},{x,…}]  消元

Eliminate[eqns,elims]       rearrange equations toeliminate the variables  elims

Reduce[{lhs₁==rhs₁,lhs₂==rhs₂,…},{x,y,…}]   化簡方程（有時比Solve更強大）

Root[f,k]   代表方程f[x]==0的第k個根

FindRoot[lhs==rhs,{x,x₀}]  尋找方程的根，從x=x0開始找

InverseFunction[f]  求反函數

InverseFunction[f,k,n]  the inverse function of the  n-argument function  f with respect to its  k_th argument

CountRoots[poly,x]  得到多項式的實根數

 

關係運算符

   等式

       unequal(also input as  x  ¹  y)

    大於

   大於等於

    小於

   小於等於

  all equal

    all unequal (distinct)

x>y>z,etc.    strictly decreasing, etc.

邏輯運算符

  非

  與

      或

Xor[p,q,…]    異或

Nand[p,q,…]    與非

Nor[p,q,…]   或非

If[p,then,else]      如果p成立，則執行then，否則執行else

Reduce[{ineq₁,ineq₂,…},x]   化簡不等式

Reduce[{ineq₁,ineq₂,…},{x₁, x₂ ,…}]    reduce a collectionof inequalities in several variables

Reduce[{expr₁,…,x₁Îdom₁,…},vars]   explicitly specify individual domains for variables

Reduce[expr,vars,dom]          reduce  eqns over the domain  dom

FindInstance[ineqs,{x₁,x₂,…}]  找到滿足不等式的一個值

FindInstance[ineqs,vars,n]       try to find  n instances

Minimize[expr,{x₁,x₂,…}]   求最小值

Minimize[{expr,cons},{x₁,x₂,…}]   minimize  expr subject to the constraints  cons

Maximize[expr,{x₁,x₂,…}]   求最大值

Maximize[{expr,cons},{x₁,x₂,…}]   maximize  expr subject to the constraints  cons

Sum[f,{i,i_min,i_max}]  求累和  the sum

Sum[f,{i,i_min,i_max,di}]   the sum with  i increasing in steps of  di

Sum[f,{i,i_min,i_max},{j,j_min,j_max}]    the nested sum 

Product[f,{i,i_min,i_max}]  求累積   

  表示無窮

Series[expr,{x,x₀,n}]    級數展開  find the power seriesexpansion of  expr about the point  x=x₀ to at most  n_th order

Series[expr,{x,x₀,n_x},{y,y₀,n_y}]    find series expansions with respect to  y, then  x

Normal[series]              truncate a power series to give an ordinary expression

RSolve[eqn,a[n],n]     解遞歸方程

RSolve[{eqn₁,eqn₂,…},{a₁[n],a₂[n],…},n]  solve a coupled system ofrecurrence equations

RSolve[eqns,a[n₁,n₂,…],{n₁,n₂,…}]    solve partial recurrenceequations

Limit[expr,x->x₀]  求極限  find the limit of  expr when  x approaches  x₀

Limit[expr,x->x₀,Direction->1]  求左極限

Limit[expr,x->x₀,Direction->-1]  求右極限

Residue[expr,{x,x₀}]   求留數   the residue of  expr when  x equals  x₀

D[f,x]   求偏導數 partial derivative

D[f,x,y,…]   multiple derivative

D[f，{x,n}]  求n階偏導數        derivative

Dt[f]   求全微分df

Dt[f,x]  求全微分

Dt[f,x,y,…]   multiple total derivative

Dt[f,x,Constants->{c₁,c₂,…}]    total derivative with  c_i constant (i.e.,  d c_i=0)

y/:Dt[y,x]=0    set  

SetAttributes[c,Constant]   define  c to be a constant in all cases

Integrate[f,x]   求f的不定積分

Integrate[f,x,y]  the multiple integral

Integrate[f,{x,x_min,x_max}]  求定積分

Integrate[f,{x,x_min,x_max},{y,y_min,y_max}]  the multiple integral

Integrate[f Boole[cond],{x,x_min,x_max},{y,y_min,y_max}]  integrate  f over the region where  cond  is  True

DSolve[eqn,y[x],x] 解微分方程  solve a differential equationfor  y[x]

DSolve[eqn,y,x]     solve a differential equationfor the function  y

DSolve[{eqn₁,eqn₂,…},{y₁,y₂,…},x]    solve a list of differentialequations

DSolve[eqn,y[x₁,x₂,…],{x₁,x₂,…}]      solve a partial differentialequation for  y[x₁,x₂,…]

DSolve[eqn,y,{x₁,x₂,…}]      solve a partial differentialequation for the function  y

LaplaceTransform[expr,t,s]  求拉普拉斯變換

InverseLaplaceTransform[expr,s,t]   求拉普拉斯反變換

LaplaceTransform[expr,{t₁,t₂,…},{s₁,s₂,…}]  the multidimensional Laplacetransform of  expr

InverseLaplaceTransform[expr,{s₁,s₂,…},{t₁,t₂,…}]   the multidimensional inverseLaplace transform of  expr

FourierTransform[expr,t,w]  傅里葉變換

InverseFourierTransform[expr,w,t]  傅里葉反變換

 

 

通過設置FourierParameters，可設置傅里葉變換的算法

FourierSinTransform[expr,t,w]  Fourier sine transform

FourierCosTransform[expr,t,w]  Fourier cosine transform

InverseFourierSinTransform[expr,w,t]  inverse Fourier sine transform

InverseFourierCosTransform[expr,w,t]   inverse Fourier cosinetransform

 

ZTransform[expr,n,z]  Z變換

InverseZTransform[expr,z,n]   Z反變換

DiracDelta[x]  衝激函數  Dirac delta function d(x)

 

HeavisideTheta[x]  類似階躍函數   Heaviside theta function q(x), equal to 0 for x<0 and 1 for x>0

DiracDelta[x₁,x₂,…]  multidimensional Dirac deltafunction

HeavisideTheta[x₁,x₂,…]  multidimensional Heavisidetheta function

DiscreteDelta[n₁,n₂,…]   discrete delta d(n₁,n₂,…)

KroneckerDelta[n₁,n₂,…]   Kronecker delta

Mean[ ]  求平均值

Mean[{{x₁,y₁,…},{x₂,y₂,…},…}]   a list of the means of the  x_i,y_i,…

Median[list]  求中間值

Max[list]  求最大值

Variance[list]  求方差

StandardDeviation[list]  求標準差

Quantile[data,q]   求分位數  gives the location before which(100q) percent of the data lie

Commonest[data]  list of the elements withhighest frequency

MeanDeviation[data]  求絕對平均差

MedianDeviation[data]  中間數平均差   median absolute deviation,median of Èx_i-medianÈ values

Covariance[v₁,v₂]  求協方差

Correlation[v₁,v₂]  求相關係數

Expectation[f[x],xélist]  求期望    (可以通過Esc dist Esc來打出é)

BinomialDistribution[n,p]  產生試驗次數爲n，每次實驗成功的概率爲p的二項分佈

BernoulliDistribution[p]  產生成功概率爲p的兩點分佈

NormalDistribution[m,s] 產生正態分佈

PDF[dist,x]   求概率密度函數

FindClusters[data]   partition  data into lists of similarelements

FindClusters[data,n]  partition  data into exactly  n lists of similar elements

Nearest[{elem₁,elem₂,…},x]  求最接近值  give the list of  elem_i to which  x is nearest

Fit[{y₁,y₂,…},{f₁,f₂,…},x]   fit the values  y_n to a linear combination offunctions  f_i

Fit[{{x₁,y₁},{x₂,y₂},…},{f₁, f₂,…},x]  fit the points  (x_n,y_n) to a linear combination ofthe  f_i

FindFit[data,form,{p₁,p₂,…},x]   find a fit to  form with parameters  p_i

Fourier[data]   數值的傅里葉變換

InverseFourier[data]   數值的傅里葉反變換

Fit[{f₁,f₂,…},{fun₁,fun₂,…},x]   曲線擬合  find a linear combination ofthe  fun_i that best fits the values  f_i

Fit[data,{fun₁,fun₂,…},{x,y,…}]   fit to a function of severalvariables

FindFit[data,form,{par₁,par₂,…},x]    search for values of the  par_i that make

 form best fit  data

 

FindFit[data,form,pars,{x,y,…}]  fit multivariate data

     finds a best fit subject tothe parameter constraints

 cons

Interpolation[{f₁,f₂,…}]   construct an approximatefunction with values  f_i at

successive integers

Interpolation[{{x₁,f₁},{x₂,f₂},…}]   求近似函數construct an approximatefunction with values  f_i at points  x_i

Fourier[{u₁,u₂,…,u_n}]  離散傅里葉變換

InverseFourier[{v₁,v₂,…,v_n}]   離散傅里葉反變換

 

 

Fourier[{{u₁₁,u₁₂,…},{u₂₁,u₂₂,…},…}]   two-dimensional discreteFourier transform

FourierDCT[list]   Fourier discrete cosinetransform of a list of real numbers

FourierDST[list]   Fourier discrete sinetransform of a list of real numbers

FourierDCT[list,m]  Fourier discrete cosinetransform of type  m

FourierDST[list,m]  Fourier discrete sinetransform of type  m

繪圖部分

Plot[f,{x,x_min,x_max}]   繪圖  plot  f as a function of  x from  x_min to  x_max

Plot[f,{x,x₀,x₁,…,x_k}]  plot  f over a series of segments,potentially breaking the curve at each of the  x_i

Show[plot,option->value]  重畫  redraw a plot with optionschanged

Show[plot₁,plot₂,…]   聯合繪圖  combine several plots

GraphicsGrid[{{plot₁,plot₂,…},…}]    draw an array of plots

InputForm[plot]       show the underlying textualdescription of the plot

Plot3D[f,{x,x_min,x_max},{y,y_min,y_max}] 畫三維圖 make a three-dimensional plotof  f as a function of thevariables  x and  y

ListPlot[{y₁,y₂,…}]  畫點圖   plot  y₁,y₂,… at  x values  1,2,…

ListPlot[{{x₁,y₁},{x₂,y₂},…}]    plot points  (x₁,y₁),…

ListLinePlot[list]  join the points with lines

ListPlot3D[{{z₁₁,z₁₂,…},{z₂₁,z₂₂,…},…}]   make a three-dimensional plotof the array of heights  z_yx

ListPlot3D[{{x₁,y₁,z₁},{x₂,y₂,z₂},…}]   make a three-dimensional plotwith heights  z_i at positions  {x_i,y_i}

ListContourPlot[array]   make a contour plot

ListDensityPlot[array]   make a density plot

ParametricPlot[{f_x,f_y},{t,t_min,t_max}] 參數繪圖

ParametricPlot[{{f_x,f_y},{g_x,g_y},…},{t,t_min,t_max}]  plot several parametric curvestogether

ParametricPlot3D[{f_x,f_y,f_z},{t,t_min,t_max}]  make a parametric plot of athree-dimensional curve

ParametricPlot3D[{f_x,f_y,f_z},{t,t_min,t_max},{u,u_min,u_max}]  make a parametric plot of athree-dimensional surface

ParametricPlot3D[{{f_x,f_y,f_z},{g_x,g_y,g_z},…},…]  plot several objects together

Play[f,{t,0,t_max}]  播放函數的聲音  play a sound with amplitude  f as a function of time  t in seconds

Play[f,{t,0,t_max},SampleRate->r]    play a sound, sampling it  r times a second