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I had to explain a complex analysis meme to a friend which meant I took 11 days of my life cramming basic complex analysis into my brain

my head horts

edit: the contour Γ is the classic semicircle around 𝑖 and all

#math #mathematics #complex analysis #integration #contour integration #residue theorem

Look, I get that there’s no nice way to have your limits of integration explicitly lay out a Hankel contour without just saying it’s a contour integral on C then explaining what C is.

But seriously. Was there a LESS intuitive way you could have written this integral? Come on....

#the Hankel contour goes from positive infinity at a distance ε from the real axis down #circles around the origin at a constant radius of ε #anticlockwise of course #then goes back up to positive infinity at a distance of ε below the real axis #so I guess it does go from positive infinity to positive infinity #but...like...#don't? do this?#why?#this is the same author who REFUSES to use the gamma function #because it would mean he'd have to write Γ(z+1)=z!#instead of writing Π(z)=z!#which is what he does #and like #I get that you're pissy that Lebesgue went and changed the original pi function and it doesn't line up with the factorial the same anymore #BUT LITERALLY EVERYONE OTHER THAN YOU USES Γ(z) FOR THAT FUNCTION #IT'S JUST YOU #mathematics #contour integration

Part 1. Green's functions for the Helmholtz (wave equation) operator in various dimensions.

[Click here for a PDF version of this post] My favorite book on mathematical physics derives the Green’s function for the 3D Helmholtz (wave equation) operator. I tried to derive the 2D Green’s function the same way and had trouble. In this series of blog posts, I’ll attempt that again, but will start with the easier 1D and 3D cases. Presuming that I don’t hit any conceptual troubles trying both…

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#acausal solution #boundary value #causal solution #Contour integration #convolution #Fourier transform #Green&039;s function #pole #residue #wave equation

A generalized Gaussian integral.

[Click here for a PDF version of this post] Here’s another problem from [1]. The point is to show that \begin{equation}\label{eqn:generalizedGaussian:20} G(x,x’,\tau) = \inv{2\pi} \int_{-\infty}^\infty e^{i k\lr{ x – x’ } } e^{-k^2 \tau} dk, \end{equation} has the value \begin{equation}\label{eqn:generalizedGaussian:40} G(x,x’,\tau) = \inv{\sqrt{4 \pi \tau} } e^{-\lr{ x – x’}^2/4 \tau…

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#complex numbers #contour #Contour integration #Gaussian #imaginary #pole #real #residue

More residue calculus: sinc squared, fractional exponent, log, pie contour

[Click here for a PDF version of this post] Sinc squared. This is problem 31(g) from [1]. Find \begin{equation}\label{eqn:sincSquared:20} I = \int_{-\infty}^\infty \frac{\sin^2 x}{x^2} dx. \end{equation} We will use the same upper half plane semicircular contour, enclosing the second order pole at the origin. This time we write \begin{equation}\label{eqn:sincSquared:40} \sin^2 x = \frac{ 1 –…

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#Contour integration #log integral #pie contour #real integrals #residue calculus

A weighted sinc function integral.

[Click here for a PDF version of this post] Here’s another real integral problem from [1] (31(f)). Find \begin{equation}\label{eqn:weightedSinc:20} I = \int_0^\infty \frac{\sin x dx}{x\lr{ x^2 + a^2 }}. \end{equation} Both Mathematica and the text state that the answer is \begin{equation}\label{eqn:weightedSinc:40} I = \frac{\pi}{2 a^2} \lr{ 1 – e^{-a}}. \end{equation} My initial attempt to…

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#Contour integration #Jordan&039;s Lemma #pole #real integral #residue #sinc

Another real integral using contour integration.

[Click here for a PDF version of this post] Here’s (31(d)) from [1]. Find \begin{equation}\label{eqn:fourPoles:20} I = \int_0^\infty \frac{dx}{1 + x^4} = \inv{2}\int_{-\infty}^\infty \frac{dx}{1 + x^4}. \end{equation} This one is easy conceptually, but a bit messy algebraically. We integrate over the contour \( C \) illustrated in fig. 1. fig. 1. Standard above the x-axis, semicircular…

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#Contour integration #real axis #residue #semicircular contour

A fun ellipse related integral.

[Click here for a PDF version of this post] Motivation. This was a problem I found on twitter ([2]) Find \begin{equation}\label{eqn:ellipicalIntegral:20} I = \int_0^\pi \frac{dx}{a^2 \cos^2 x + b^2 \sin^2 x}. \end{equation} I posted my solution there (as a screenshot), but had a sign wrong. Here’s a correction. Solution. Let’s first assume we aren’t interested in the \( a^2 = b^2 \), nor either…

#Contour integration #ellipse #integral #residue

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