#contour integral

[qedmirage]

Wanna know a secret?

This. This is how you divide by zero.

[peeterjoot]

Green's function for the 3D wave equation.

[Click here for a PDF version of this and related posts] We’ve now evaluated the 1D Green’s function and 2D Green’s function for the wave equation. For the sake of completeness, now let’s evaluate the Green’s function for the 3D wave equation operator. Again with \( \Br = \Bx – \Bx’, \tau = t – t’ \) we want the \( \epsilon \rightarrow 0 \) limit…

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[peeterjoot]

Derivation of the 2D Green's function for the wave equation operator.

[Click here for a PDF version of this post] While it was difficult to attempt to verify the 2D Green’s function, it actually turns out to be fairly easy to derive it, provided we pick an alternate pole displacement from the 1D evaluation to make our lives easier. With \( \Br = \Bx – \Bx’ \), and \( \tau = t – t’ \), and \( \epsilon > 0 \), we can…

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[peeterjoot]

Correcting the errors: Green's functions for the 1D Helmholtz and Laplacian operators.

[Click here for a PDF version of this post, and others in this series] The following recent posts explored 1D Green’s functions for the Helmholtz and Laplacian operators.  There was a sign error (wrong residue sign for a negatively oriented contour) that I made near the beginning that caused a lot of trouble.  Having found the error, I’ve now reworked all that exploratory content into a more…

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[peeterjoot]

Part 2. 3D Green's function for the Helmholtz (wave equation) operator

[Click here for a PDF version of this post] This is a continuation of yesterday’s post on wave function Green’s functions. We derived the 1D Green’s function, now it’s time for the 3D. Next up after this will be the 2D Green’s function, which looks harder to evaluate than the 1D or 3D versions. 3D Green’s function. The 3D Green’s function that we wish to try to evaluate…

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[peeterjoot]

A Green's function solution to falling with resistance problem.

[Click here for a PDF version of this post] Motivation. In a fun twitter/x post, we have a Green’s function solution to a constant acceleration problem with drag. The post is meant to be a joke, as the stated problem is: “A boy drops a ball from a height \( h \). What is the speed of the ball when it reaches the floor (no drag)?” The joke is that nobody would solve this problem using Green’s…

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[peeterjoot]

A PV integral using contour integration.

[Click here for a PDF version of this post] Here’s the second last real-integral sub-problem from [1], problem 31(j). Find \begin{equation}\label{eqn:oscillatorKernel:20} I = P \int_{-\infty}^\infty \inv{ \lr{ \omega’ – \omega_0 }^2 + a^2 } \inv{ \omega’ – \omega } d\omega’. \end{equation} Our poles are sitting at \( \omega \), and \begin{equation}\label{eqn:oscillatorKernel:80} \alpha, \beta =…

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[peeterjoot]

A contour integral with a third order pole.

[Click here for a PDF version of this post] Here’s problem 31(e) from [1]. Find \begin{equation}\label{eqn:thirdOrderPole:20} I = \int_0^\infty \frac{x^2 dx}{\lr{ a^2 + x^2 }^3 }. \end{equation} Again, we use the contour \( C \) illustrated in fig. 1 fig. 1. Standard above the x-axis, semicircular contour. Along the infinite semicircle, with \( z = R e^{i\theta}…

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