In[]:=
CompoundExpression[
]
​​deploy
Thu 18 May 2023 19:05:53
In[]:=
ilaplaceMellin[expr_]:=Block[{repl,ilap,mellin,a1,a2,a3,a4,a},repl={ArcTan[a1_]->ArcTan[a1/a],Log[a1_]->Log[a1/a],Hypergeometric2F1[a1_,a2_,a3_,a4_]->Hypergeometric2F1[a1,a2,a3,a4/a]};​​mellin=MellinTransform[expr/.repl,a,q]//FunctionExpand;​​ilap=InverseLaplaceTransform[mellin,s,t];​​FullSimplify[InverseMellinTransform[ilap,q,a]/.a->1]];​​F=
ArcTan
2
s

2
s
;​​G=
ArcTan
2
s

2
s
1+
1
4
-2+
2
s
ArcTan
2
s

;​​Print"f: ",f=ilaplaceMellin
ArcTan
2
s

2
s
​​Print["
-1
L
​(G/F)=",ilaplaceMellin[F/G]]
f:
π
2
Erf[
2
t
]
2
t
-1
L
​(G/F)=
DiracDelta[t]
2
+
1
16
4
-2t

t
-
2π
Erf[
2
t
]
3/2
t
In[]:=
ilaplaceMellin[F/G]
Out[]=
DiracDelta[t]
2
+
1
16
4
-2t

t
-
2π
Erf[
2
t
]
3/2
t
In[]:=
maxT=10000;​​sf={"Log","Log"};​​positions=Floor[Exp[#]]&/@Subdivide[Log[1],Log[maxT],10];​​sgdLosses=Tablet,
InverseLaplaceTransform[F,s,t]
InverseLaplaceTransform[G,s,t]
,{t,N[positions]};​​observedPlot=ListPlot[sgdLosses,ScalingFunctions->sf];​​Show[observedPlot]​​
Out[]=