In[]:=
CompoundExpression[
]
​​deploy
Sat 2 Dec 2023 23:52:40
In[]:=
With[{a=1,b=2},​​points=ListPlot[{Labeled[{a,1},"a"],Labeled[{b,1},"b"]},FillingAxis,FillingStyleDashed];​​theta=HeavisideTheta[y-a]-HeavisideTheta[y-b];​​plot=Plot[theta,{y,0,3},Filling->Axis,PlotRangePadding->.2];​​Show[plot,points]​​]
Out[]=
In[]:=
points
Out[]=
In[]:=
g[x_]=
1
2
(1+x)
;​​gi=InverseFunction[g];​​dgi=D[gi[y],y];​​a=g[0];​​b=g[∞];​​theta=HeavisideTheta[y-b]-HeavisideTheta[y-a];
InverseFunction
:Inverse functions are being used. Values may be lost for multivalued inverses.
In[]:=
LaplaceTransform[D[gi[y],y]*y*theta,y,t]//FullSimplify
Out[]=
π
Erf[
t
]
2
t
In[]:=
Integrate[g[x]Exp[-tg[x]],{x,0,∞}]
Out[]=
π
Erf[
t
]
2
t
In[]:=
Plot
π
-Erf[
t
]+Erf[
2
t
]
2
t
,
π
Erf[
t
]
2
t
,{t,1,10}
Out[]=
Asymptotic
In[]:=
Solve[g[0]==Infinity,x]
Out[]=
{}
In[]:=
Plot[g[x],{x,0,10}]
Out[]=
In[]:=
g[0]
Out[]=
1
In[]:=
continuousPlot2=Plot[sol2,{s,0,maxStep},PlotRangeAll];​​Show[discretePlot,continuousPlot2]
Out[]=
exact
In[]:=
IntegrateExp-2s
1
2
(i)
,{i,0,n},Assumptions{n>1,s>1}//FullSimplify
Out[]=
-
2s
2
n

n-
2π
s
Erfc
2
s
n

In[]:=
Block{n},​​IntegrateExp-2s
1
2
i
,{i,0,nn},Assumptions{nn>1,s>1}​​
Out[]=
-
2s
2
nn

nn-
2π
s
Erfc
2
s
nn

In[]:=
With{nn=10},​​ShowdiscretePlot,Plot
-
2s
2
nn

nn-
2π
s
Erfc
2
s
nn
,{s,1,100}​​
Out[]=
exact
In[]:=
LeafCount[-E^-t]
Out[]=
7