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In[]:=
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Tue 7 Sep 2021 17:45:26

Solution for increasing g(x)

Invert f(s) obtained from g(x)=x+1 using solution in math.SE post
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$Assumptions={s>0,y>1,t>0};​​g[x_]=x+1;​​f[s_]=Integrate[g[x]Exp[-sg[x]],{x,0,Infinity}];​​​​Plot[{g[x],f[x]},{x,0,2},PlotLegends->{g,f}]
Out[]=
0.5
1.0
1.5
2.0
1
2
3
4
5
6
7
g
f
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F[t_]=Integrate[f[s],{s,t,Infinity}]
Out[]=
-t
t
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InverseLaplaceTransform[F[t],t,s]​​gi[y_]=Integrate[%,{s,0,y}]
Out[]=
HeavisideTheta[-1+s]
Out[]=
-1+y
In[]:=
ParametricPlot[{{y,f[y]},{y,g[y]},{gi[y],y}},{y,0,10},PlotStyle->{Automatic,Bold,Dashed},PlotLegends{"f","g","inferred g"}]
Out[]=
2
4
6
8
10
5
10
15
20
f
g
inferred g
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Print["g is ",SolveValues[gi[z]==x,z]//First]
g is 1+x

Other examples

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summarize[gg_]:=(​​g[x_]=gg;​​f[x_]=Integrate[g[x]Exp[-sg[x]],{x,0,Infinity}];​​Print["g(x)=",g[x]];​​Print["f(s)=",f[s]];​​Print["ℒ(g)=",LaplaceTransform[g[x],x,y]];​​Print["ℒ(f)=",LaplaceTransform[f[s],s,t]];​​Print["
-1
ℒ
​(g)=",InverseLaplaceTransform[g[x],x,y]];​​Print["
-1
ℒ
​(f)=",InverseLaplaceTransform[f[s],s,t]];​​Print["ℒ(g')=",LaplaceTransform[D[g[x],x],x,t]//TraditionalForm];​​)
In[]:=
summarize[1+x]
g(x)=1+x
f(s)=
-s
(1+s)
2
s
ℒ(g)=
1
2
y
+
1
y
ℒ(f)=LaplaceTransform
1+s
2
s
,s,1+t
-1
ℒ
​(g)=DiracDelta[y]+
′
DiracDelta
[y]
-1
ℒ
​(f)=tHeavisideTheta[-1+t]
ℒ(g')=
1
t
In[]:=
summarize
1
2
(1+x)
g(x)=
1
2
(1+x)
f(s)=
π
Erf[
s
]
2
s
ℒ(g)=1+
y
yExpIntegralEi[-y]
ℒ(f)=
ArcTan
1
t
t
-1
ℒ
​(g)=
-y
y
-1
ℒ
​(f)=
1-HeavisideTheta[-1+t]
2
t
ℒ(g')=
t
2
t
Ei(-t)+t-1
In[]:=
summarize[Exp[-x]]
g(x)=
-x
f(s)=
1-Cosh[s]+Sinh[s]
s
ℒ(g)=
1
1+y
ℒ(f)=Log1+
1
t
-1
ℒ
​(g)=DiracDelta[-1+y]
Limit
:Warning: Assumptions that involve the limit variable are ignored.
Limit
:Warning: Assumptions that involve the limit variable are ignored.
Limit
:Warning: Assumptions that involve the limit variable are ignored.
General
:Further output of Limit::alimv will be suppressed during this calculation.
-1
ℒ
​(f)=1-HeavisideTheta[-1+t]
ℒ(g')=-
1
t+1
In[]:=
summarize[
2
Log[x]
]
g(x)=
2
Log[x]
f(s)=
1
4
s
π
(1+2s)
4
5/2
s
if Re[s]>0
ℒ(g)=
6
2
EulerGamma
+
2
π
+6Log[y](2EulerGamma+Log[y])
6y
ℒ(f)=-π
t
Cos[
t
]
-1
ℒ
​(g)=InverseLaplaceTransform[
2
Log[x]
,x,y]
Out[]=
$Aborted
In[]:=
summarize[Log[x+1]]
g(x)=Log[1+x]
f(s)=
1
2
(-1+s)
ℒ(g)=-
y
ExpIntegralEi[-y]
y
ℒ(f)=-1+
-t
t(-π+ExpIntegralEi[t])
-1
ℒ
​(g)=InverseLaplaceTransform[Log[1+x],x,y]
-1
ℒ
​(f)=
t
t
ℒ(g')=-
t
Ei(-t)
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