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Calculus and Beyond Homework Help
Does a double principal-value integral exist?
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[QUOTE="aheight, post: 6467461, member: 600309"] [B]Homework Statement:[/B] Does a double-integral with a principal-value converge? [B]Relevant Equations:[/B] Not sure Encountered this integral and I believe it converges by studying it numerically but not sure and was wondering how might I show it converges or diverges? Surely there must be a way. $$ \text{P.V.}\int_0^{\infty}\int_0^{\infty} \frac{\text{sinc}^2(1+\phi)v e^{-v}}{(e^{v/2}-1)(\phi-v)}dvd\phi $$ where ##\text{sinc}(x)=\frac{\sin(x)}{x}##. This is my work so far: Note the inner iterated integral has two singular points: 0 and ##\phi## so I can write: $$ \begin{align*} &\text{P.V.}\int_0^{\infty}\int_0^{\infty} \frac{\text{sinc}^2(1+\phi)v e^{-v}}{(e^{v/2}-1)(\phi-v)}dvd\phi\\ &=\int_0^{\infty}\biggr\{\text{P.V}\int_0^{\infty}\frac{\text{sinc}^2(1+\phi)v e^{-v}}{(e^{v/2}-1)(\phi-v)}dv\biggr\} d\phi \end{align*} $$ so now, how about I let: $$ g(\phi)=\text{P.V}\int_0^{\infty}\frac{\text{sinc}^2(1+\phi)v e^{-v}}{(e^{v/2}-1)(\phi-v)}dv $$ Now, just for starters, consider ##g(5)##: $$ g(5)=\text{P.V}\int_0^{\infty}\frac{\text{sinc}^2(6)v e^{-v}}{(e^{v/2}-1)(5-v)}dv $$ with singular points at 0 and 5. Again for starters, how about I just try to numerically integrate g(5) in the principal-valued sense in Mathematica: [CODE title="Mathematica"]In[40]:= myf2[v_,\[Phi]_]:=(v Exp[-v])/((Exp[v/2]-1)(\[Phi]-v)) NIntegrate[Sinc[6]^2myf2[v,5],{v,0,0,5,\[Infinity]},Method->"PrincipalValue"] Out[41]= 0.000858798[/CODE] and supprisingly it at least converges numerically in the principal-valued sense. Next, how about I try to set up a Riemann-sum type rule to numerically integrate ##g(\phi)## over some range say ##0.0001\leq \phi\leq 15##: [CODE title="Mathematica"] myf2[v_, \[Phi]_] := (v Exp[-v])/((Exp[v/2] - 1) (\[Phi] - v)) sum = 0; deltaPhi = 1/100; myDoubleSum = Table[ sum += (deltaPhi NIntegrate[ Sinc[1 + thePhi]^2 myf2[v, thePhi], {v, 0, 0, thePhi, \[Infinity]}, Method -> "PrincipalValue"]); {thePhi, sum}, {thePhi, 0.0001, 15, deltaPhi}]; [/CODE] And this results in the plot below which seems to be converging at least in this range so it looks like it may converge. [ATTACH type="full" alt="doublepv.jpg"]279590[/ATTACH] [/QUOTE]
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Does a double principal-value integral exist?
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