Emergence of Cauchy Principal valueI have a problem solving differential equation, where I think there should be a Cauchy Principal value involved, but I do not see how it it should emerge. Let's say we have a differential equation:iC˙1(t)=∫0∞dω ν(ω)ei(ω10-ω)t,where all the variables are real and positive. If we now solve the differential equation, it needs only to carry out the time integraliC1(t)=∫dt∫0∞dω ν(ω)ei(ω10-ω)t=∫0∞dω ν(ω)i(ω10-ω)ei(ω10-ω)t.So the problem is that if the last integral is not taken as principal value, it will give&nbsp;∞, so I would rather need it to beP∫0∞dω ν(ω)i(ω10-ω)ei(ω10-ω)twhere&nbsp;P&nbsp;is the principal value. Would anyone have any idea where the principal value would emerge from, if at all? I was thinking maybe it has something to do when swapping the order of integrals in order to carry out the time integral first, but I am not sure.

Question

Emergence of Cauchy Principal valueI have a problem solving differential equation, where I think there should be a Cauchy Principal value involved, but I do not see how it it should emerge. Let&#039;s say we have a differential equation:iC˙1(t)=∫0∞dω ν(ω)ei(ω10-ω)t,where all the variables are real and positive. If we now solve the differential equation, it needs only to carry out the time integraliC1(t)=∫dt∫0∞dω ν(ω)ei(ω10-ω)t=∫0∞dω ν(ω)i(ω10-ω)ei(ω10-ω)t.So the problem is that if the last integral is not taken as principal value, it will give&amp;nbsp;∞, so I would rather need it to beP∫0∞dω ν(ω)i(ω10-ω)ei(ω10-ω)twhere&amp;nbsp;P&amp;nbsp;is the principal value. Would anyone have any idea where the principal value would emerge from, if at all? I was thinking maybe it has something to do when swapping the order of integrals in order to carry out the time integral first, but I am not sure.

Ashton Conrad · Accepted Answer

Explanation:Yes, your observation is precisely that Fubini does not apply, because one of the iterated integrals is (sometimes) infinity. But you can cut off a small ball around ω10&amp;gt;0 before interchanging the integrals:PSKbegin{align} int_0^t int_0^infty v(omega)e^{i(omega_{10}-omega)t}domega dt&amp;amp;= lim_{epsilonto 0}int_0^tint_{omegain[0,infty),|omega-omega_{10}|&amp;gt;epsilon} v(omega)e^{i(omega_{10}-omega)t}domega dt&amp;amp;= lim_{epsilonto 0}int_{omegain[0,infty),|omega- omega_{10}|&amp;gt;epsilon} int_0^tv(omega)e^{i(omega_{10}-omega)t}domega dt&amp;amp;= lim_{epsilonto 0}int_{omegain[0,infty),|omega-omega|&amp;gt;epsilon} frac{v(omega)e^{i(omega_{10}-omega)t}}{i(omega_{10}-omega)}domega &amp;amp;=:mathcal P!!!int_0^infty frac{v(omega)e^{i(omega_{10}-omega)t}}{i(omega_{10}-omega)}domegaZSKThe above works only when ω10≠0ZKS.Also,ifPSKv(ω10)=0 and is C1 then the above still works but then the integral does exist, so P!!∫0∞=∫0∞.

Emergence of Cauchy Principal value I have a problem

Answered question

Answer & Explanation

New Questions in Calculus 2