Wolfram Mathematical Functions | Things to Try

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.
Define, compute and visualize. Symbolic and numerical evaluation, visualization and asymptotic expansions of a large collection of mathematical functions—extensively documented and tightly integrated with all areas of Wolfram Language.

Elementary Functions

Factor a polynomial:
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In[]:=
Factor[
15
x
-1]
Compute a Gröbner basis for polynomials:
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In[]:=
GroebnerBasis[{
2
x
-2
2
y
,xy-3},{x,y}]
Verify identities between functions:
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Simplify[Sin[x+y]==Sin[x]Cos[y]+Cos[x]Sin[y]]
Compute a series expansion:
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Series[ArcTan[x],{x,0,10}]
Solve a simple growth model whose solution is expressible as elementary functions:
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DSolveValue[{x'[t]-rx[t]==0,x[3]==10},x[t],t]
Interactively explore the effect of the growth parameter:
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In[]:=
Manipulate[Plot[10
-3r+rt

,{t,0,3},PlotRange->{0,10}],{r,1,5}]

Special Functions

Compute derivatives of a hypergeometric function:
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In[]:=
D[Hypergeometric2F1[a,b,c,z],{z,n}]
Compute contour integrals of special functions:
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In[]:=
ContourIntegrate[Hypergeometric2F1[2,3,4,z],z∈Circle[{0,0},2]]
Visualize an elliptic function:
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ComplexPlot3D[JacobiSN[z,1/2],{z,-4-4I,4+4I},PlotLegends->Automatic]
Compare the behavior of Bessel functions:
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ReImPlot[{BesselJ[0,x],BesselY[0,x],AiryAi[x]},{x,-5,5},PlotLegends->"ReImExpressions"]

Piecewise and Generalized Functions

Define and use piecewise functions as part of computations:
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[x_]:=Piecewise[{{-x+1,x<0},{x,0<x<1},{3x^2-2,True}}]​​ℊ[x_]:=
x
∫
0
[t]t​​Plot[{[x],ℊ[x]},{x,-2,2},PlotLegends->"Expressions"]​​
Express the fundamental solution of the Klein–Gordon operator (
∂
tt
-
2
∇

x
+m
) in terms of generalized functions:
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In[]:=
[t_,x_List]:=1/(2π)HeavisideTheta[t]DiracDelta[t^2-x.x]-m/(4π)HeavisideTheta[t-Sqrt[x.x]]BesselJ[1,mSqrt[t^2-x.x]]/Sqrt[t^2-x.x]​​[t,{x,0,0}]//TraditionalForm
Plot the function, which is nonzero only for
2
t
>

x
·

x
:
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DensityPlot-[t,{x,0,0}]/.m->1,{x,-3,3.02},{t,-3,3.01},Exclusions{{
2
t

2
x
,t≥0}},FrameLabel(Style[TraditionalForm[#1],16]&)/@{x,t},


Integer Functions

Compare the average number of divisors per integer to its asymptotic value:
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In[]:=
Show​​ListPlotTable
1
n
n
∑
i=1
DivisorSigma[0,i],{n,100},​​Plot[Log[n]+2EulerGamma-1,{n,1,100},PlotStyle->ColorData[97,2]]​​
Plot the number of primes less than or equal to
x
as compared with functions that estimate
π(x)
:
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In[]:=
Plot[{PrimePi[x],x/Log[x],LogIntegral[x],RiemannR[x]},{x,1.5,100},PlotLabels->"Expressions"]

Compute Function Properties

Find the period of a function:
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FunctionPeriod[Sin[ωx],x]
Test for injectivity:
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In[]:=
FunctionInjective[
3
x
+ax+b,x]
Test for surjectivity:
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In[]:=
FunctionSurjective[
3
x
+ax+b,x]
Test for bijectivity:
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In[]:=
FunctionBijective[
3
x
+ax+b,x]
Find poles of a function:
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In[]:=
FunctionPoles[Gamma[z],z]
Test for analyticity:
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In[]:=
FunctionAnalytic[Gamma[z],z,Complexes]
Test whether a function is meromorphic:
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In[]:=
FunctionMeromorphic[Gamma[z],z]

Compute Exact Symbolic Results

Compute integrals of rational functions:
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In[]:=
1
∫
-1
2
4
x
+1
-
2
x
x
Compute the Fourier transform of a function:
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In[]:=
FourierTransform[
-Abs[t]

,t,ω]
Compute the Mellin transform of a multivariate rational function:
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In[]:=
MellinTransform
1
x+y^2+1
,{x,y},{s,t}
Solve a differential equation exactly:
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In[]:=
DSolveValue[{x''[t]+Sin[x[t]]==0,x[0]==1,x'[0]==0},x[t],t]

Compute Numeric Results

Evaluate functions to specified numeric precision:
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In[]:=
N[JacobiSN[1,1/3],50]
Compute integrals numerically:
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In[]:=
Γintegral[z_?NumberQ]:=NIntegrate[t^(z-1)Exp[-t],{t,0,∞}]​​{Γintegral[2+3],N[Gamma[2+3]]}//Column
Include uncertainty in expressions:
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In[]:=
ArcCos[Around[u,.1]+IAround[v,.1]]
Visualize a function with uncertainty:
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In[]:=
ListLinePlot[Table[{x,Sin[Around[x,1/4]]},{x,0,2Pi,.1}],IntervalMarkers->"Bands"]

Compute Asymptotic Relationships

Compute an asymptotic approximation to
n!
:
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In[]:=
DiscreteAsymptotic[n!,n->∞]
Check that the approximation and expression are equivalent:
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In[]:=
AsymptoticEquivalentn!,
-n

1
2
+n
n
2π
,n->∞