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MathGroup Archive 2007

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curious results form Integrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73730] curious results form Integrate
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Tue, 27 Feb 2007 05:39:42 -0500 (EST)

Hello to all.

$Version
"5.2 for Microsoft Windows (June 20, 2005)"


Integrate[Sin[x + x0]/(x + x0), {x, -Infinity, Infinity}]
Pi
(*correct regrardless x0; there is an integrable singularity at x=-
x0*)

Limit[Sin[x + x0]/(x + x0), x -> -x0]
1

Integrate[Sin[x + x0]/(x + x0), {x, -Infinity, x0, Infinity}]
Expand[%]
Pi/2 + (1/2)*(Pi - 2*SinIntegral[2*x0]) + SinIntegral[2*x0]
Pi

(NIntegrate[Sin[x + #1]/(x + #1), {x, -Infinity, Infinity}, Method ->
Oscillatory] & ) /@ Range[-3, 3]
{3.14159,3.14159,3.14159,3.14159,3.14159,3.14159,3.14159}


HOWEVER

Integrate[Sin[x + d]/(x + d), {x, -Infinity, 0}]
Integrate[Sin[x + d]/(x + d), {x, 0, Infinity}]

If[Im[d] != 0 || Re[d] <= 0, Pi/2 + SinIntegral[d], Integrate[Sin[d -
x]/(d - x), {x, 0, Infinity}, Assumptions -> d > 0]]
If[Im[d] != 0 || Re[d] >= 0, (1/2)*(Pi - 2*SinIntegral[d]),
Integrate[Sin[d + x]/(d + x), {x, 0, Infinity},
   Assumptions -> d < 0]]

I don't understand the presence of the If's statements.

Note also that some of the integrals below stay unevaluated!

(Integrate[Sin[x + #1]/(x + #1), {x, -Infinity, Infinity}] & ) /@
Range[-3, 3]
(Integrate[Sin[x + #1]/(x + #1), {x, -Infinity, 0}] & ) /@ Range[-3,
3]
(Integrate[Sin[x + #1]/(x + #1), {x, 0, Infinity}] & ) /@ Range[-3, 3]

{Integrate[-(Sin[3 - x]/(-3 + x)), {x, -Infinity, Infinity}],
Integrate[-(Sin[2 - x]/(-2 + x)), {x, -Infinity, Infinity}],
  Integrate[-(Sin[1 - x]/(-1 + x)), {x, -Infinity, Infinity}], Pi,
Integrate[Sin[1 + x]/(1 + x), {x, -Infinity, Infinity}],
  Integrate[Sin[2 + x]/(2 + x), {x, -Infinity, Infinity}],
Integrate[Sin[3 + x]/(3 + x), {x, -Infinity, Infinity}]}

{(1/2)*(Pi - 2*SinIntegral[3]), (1/2)*(Pi - 2*SinIntegral[2]),
(1/2)*(Pi - 2*SinIntegral[1]), Pi/2,
  Integrate[Sin[1 + x]/(1 + x), {x, -Infinity, 0}], Integrate[Sin[2 +
x]/(2 + x), {x, -Infinity, 0}],
  Integrate[Sin[3 + x]/(3 + x), {x, -Infinity, 0}]}

{Integrate[-(Sin[3 - x]/(-3 + x)), {x, 0, Infinity}], Integrate[-
(Sin[2 - x]/(-2 + x)), {x, 0, Infinity}],
  Integrate[-(Sin[1 - x]/(-1 + x)), {x, 0, Infinity}], Pi/2, (1/2)*(Pi
- 2*SinIntegral[1]), (1/2)*(Pi - 2*SinIntegral[2]),
  (1/2)*(Pi - 2*SinIntegral[3])}


Of course NIntegrate works correctly:

(NIntegrate[Sin[x + #1]/(x + #1), {x, -Infinity, 0}, Method ->
Oscillatory] & ) /@ Range[-3, 3]
(NIntegrate[Sin[x + #1]/(x + #1), {x, 0, Infinity}, Method ->
Oscillatory] & ) /@ Range[-3, 3]
% + %%

{-0.277856,-0.0346167,0.624713,1.5708,2.51688,3.17621,3.41945}
{3.41945,3.17621,2.51688,1.5708,0.624713,-0.0346167,-0.277856}
{3.14159,3.14159,3.14159,3.14159,3.14159,3.14159,3.14159}



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