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I have a little question about the numerical integration of a function that includes a Dirac delta. I have the following function:

f[x_,y_,z_]=Cot[x + z] Cot[x - y z] (-Cot[x - y z] Csc[x + z]^2 + y Cot[x + z] Csc[x - y z]^2)

Then I have to compute the following integral:

NIntegrate[DiracDelta[y - 1] f[x, y, z], {x, 0.1, 0.2}, {y, -Infinity,Infinity},{z, 0, x + y},AccuracyGoal->5]

The idea is: evaluate the symbolic integral in the variable 'z', then evaluate the resulting function of 'x' and 'y' for 'y=1' (this is perfermed by the integration of the Dirac delta), then compute the integral on the variable 'x'.

With numerical integration I obtaine the result of '0.',but it is not correct. I have tried with non-numerical integral:

Integrate[DiracDelta[y - 1] f[x, y, z], {x, 0.1, 0.2}, {y, -Infinity,Infinity},{z, 0, x + y}]

and I obaine the result of '-142.527'.

I need this numeric integral because, the function f[x,y,z], might not admit a primitive closed function.

Thanks for any tips and helps!

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    $\begingroup$ Since z goes past x it looks like it diverges? $\endgroup$ Jan 3, 2022 at 20:56
  • $\begingroup$ I don't think it diverges, infact the analitic integral converges at '-142.527'. I think the problem is in the integration of the Dirac delta function... $\endgroup$ Jan 3, 2022 at 21:10
  • $\begingroup$ I can use a simplier function like 'f[x_,y_,z_]=x y z', the analitic integral gives the correct result, the nuemrical integral gives me '0' $\endgroup$ Jan 3, 2022 at 21:13
  • $\begingroup$ @Lorenzo Your Integrate expression returns the input with an error: "Integral of Cos[2 x] Csc[x-z]^2 Csc[x+z]^2 DiracDelta[-1+y] Sin[2 z] does not converge on {0, x+y}." on MMA 12.0 (Win10-64). How did you get that -142 numerical value? $\endgroup$
    – MarcoB
    Jan 3, 2022 at 21:36
  • $\begingroup$ I am sorry, you all are right; I wrote the wrong function. I'm going to edit my question. $\endgroup$ Jan 3, 2022 at 21:51

1 Answer 1

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Mathematica Can compute this integral symbolically:

$Version
(* "13.0.0 for Microsoft Windows (64-bit) (December 3, 2021)"*)

f[x_, y_, z_] = Cot[x + z] Cot[
x - y z] (-Cot[x - y z] Csc[x + z]^2 + y Cot[x + z] Csc[x - y z]^2);

Z = Integrate[f[x, y, z], {z, 0, x + y}, Assumptions -> {x > 0, y \[Element] Reals}]

(*Long time computation about few minutes*)

 (*1/2 (-Cos[2 x] Csc[x]^4 + 
 Cos[3 x + y - y (x + y)] Cos[x + y + y (x + y)] Csc[2 x + y]^2 Csc[
 x - y (x + y)]^2)*)

Y = Integrate[DiracDelta[y - 1] Z[[1]], {y, -Infinity, Infinity}, Assumptions -> x > 0]
(*-(1/2) Cos[2 x] (Csc[x]^4 - Cos[2 + 2 x] Csc[1]^2 Csc[1 + 2 x]^2)*)

X = Integrate[Y, {x, 1/10, 2/10}]
(*-(1/6) (1 + 3 Cos[1/5] + 3 Cos[2/5]) Csc[1/5]^3 + 
1/40 Csc[1]^2 (-2 + 5 Cos[2] (Cot[6/5] - Cot[7/5]) + 
Csc[1] (-5 Csc[6/5] Sin[1/5] + 5 Csc[7/5] Sin[2/5]))*)

X//N
(*-142.527*)
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  • $\begingroup$ The integral Integrate[DiracDelta[y - 1] Z[[1]], {y, -Infinity, Infinity}, Assumptions -> x > 0] makes no sense in math. That was discussed here many times. Both Encyclopedia of Mathematics and W. Rudi, Functional analysis say nothing about the definite integrals of distributions. $\endgroup$
    – user64494
    Jan 4, 2022 at 13:27
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    $\begingroup$ @user64494 ,The integral makes sense in math. That was discussed here many times. $\endgroup$ Jan 4, 2022 at 13:30
  • $\begingroup$ Can you give a serious math reference instead of your ungrounded claim? TIA. $\endgroup$
    – user64494
    Jan 4, 2022 at 13:33
  • $\begingroup$ Thanks for the answer, but there can be cases of functions f[x,y,z] that can not be integrated symbolically, and I can't use your prescription... For example if the function is 'f[x_,y_,z_]=Exp[-z^2 Exp[-z^2]]' , the symbolic integrate doesen't work... $\endgroup$ Jan 4, 2022 at 13:37
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    $\begingroup$ And @user64494, the integral makes perfect sense. It is an example of the first integral presented in the encyclopediaofmath.org/wiki/Delta-function paper you referenced me that verified one of my previous answers. $\endgroup$
    – Bill Watts
    Jan 5, 2022 at 6:48

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