# Question:Vector Integration in Maple 2018 (compared to Maple 2017)

## Question:Vector Integration in Maple 2018 (compared to Maple 2017)

Maple 2018

Is it possible to get the following integral involving vectors done in Maple 2018, but handle the singularity when p1=p3?

Note that the output of p3 and p1 are slightly different, probably because the earlier version of the worksheet was in Maple 2017?

(Note: Earlier version - was due to help received from Dr. Edgardo Cheb-Terrab, using Maple 2017).

```with(Physics[Vectors]);
r_ := _i*x+_j*y+_k*z;
r_ := _i x + _j y + _k z
p__1_ := _i*`p__1x `+_j*`p__1y `+_k*`p__1z `;
p__1_ := _i p__1x  + _j p__1y  + _k p__1z
p__3_ := _i*`p__3x `+_j*`p__3y `+_k*`p__3z `;
p__3_ := _i p__3x  + _j p__3y  + _k p__3z
Expression to integrate wrt p3
'exp(i*(p__1_ . r_))/((p__3_ - p__1_)^2)'
exp(I Physics:-Vectors:-.(p__1_, r_))
-------------------------------------
2
Physics:-Vectors:-+(p__3_, -p__1_)
Delaying the evaluation above, if evaluation is allowed, the integrand is
exp(I*(p__1_ . r_))/(p__3_-p__1_)^2;
(exp(I (p__1x  x + p__1y  y + p__1z  z)))/((_i (-p__1x  + p__3x )

+ _j (-p__1y  + p__3y ) + _k (-p__1z  + p__3z ))^2)
T i is the imaginary unit
interface(imaginaryunit = I);
I
Because (4), has the value of i before being the imaginary unit, input the integrand again
'exp(i*(p__1_ . r_))/((p__3_ - p__1_)^2)'
exp(I Physics:-Vectors:-.(p__1_, r_))
-------------------------------------
2
Physics:-Vectors:-+(p__3_, -p__1_)

Now on the integration
Int(exp(I*(p__1_ . r_))/(p__3_-p__1_)^2, [`p__3x ` = -infinity .. infinity, `p__3y ` = -infinity .. infinity, `p__3y ` = -infinity .. infinity]);
/infinity   /infinity   /infinity
|           |           |
|           |           |          (exp(I (p__1x  x + p__1y  y
|           |           |
/-infinity  /-infinity  / -infinity

+ p__1z  z)))/((_i (-p__1x  + p__3x ) + _j (-p__1y  + p__3y )

+ _k (-p__1z  + p__3z ))^2) dp__3x  dp__3y  dp__3y
value(Int(exp(I*(`p__1x `*x+`p__1y `*y+`p__1z `*z))/(_i*(-`p__1x `+`p__3x `)+_j*(-`p__1y `+`p__3y `)+_k*(-`p__1z `+`p__3z `))^2, [`p__3x ` = -infinity .. infinity, `p__3y ` = -infinity .. infinity, `p__3y ` = -infinity .. infinity]));
/         /[
exp(I (p__1x  x + p__1y  y + p__1z  z)) |PIECEWISE|[infinity,
\         \[

Im(_i p__1x  + _j p__1y  + _k p__1z  - _k p__3z )    ]
------------------------------------------------- = 0],
_i                            ]

\\
[0, otherwise]|| infinity
//
In the above, there is the product of three Dirac delta functions, that can be represented as a single 3D Dirac delta
combine(exp(I*(`p__1x `*x+`p__1y `*y+`p__1z `*z))*piecewise(Im(_i*`p__1x `+_j*`p__1y `+_k*`p__1z `-_k*`p__3z `)/_i = 0, infinity, 0)*infinity);
/         /[
exp(I (p__1x  x + p__1y  y + p__1z  z)) |PIECEWISE|[infinity,
\         \[

Im(_i p__1x  + _j p__1y  + _k p__1z  - _k p__3z )    ]
------------------------------------------------- = 0],
_i                            ]

\\
[0, otherwise]|| infinity
//

``` ﻿