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Complex Analysis is a particularly useful mathematical tool to have in your toolbox. It can significantly simplify complicated integration. It often makes pretty ingenious use of complex numbers in order solve otherwise intractable problems. The focus of this tutorial series is to solve the ‘Planck Integral’ of Blackbody Radiation. I cover what I feel is all the necessary mathematics in order to fully understand the integral’s amazingly simply looking solution.

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Video #

Video Tutorial Title



Cauchy Riemann Equations

Path independent differentiation

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Derivation of Green’s Theorem

(2 Methods)

For path independent integration

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Derivation of the Divergence Theorem

Fundamental to vector calculus

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Relationship Between Green’s Theorem and Divergence Theorem

Green’s Theorem is a 2D Divergence Theorem

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Cauchy Integral Theorem

Closed contour integrals of analytic functions

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Differential Arc Length Formula

 A quick derivation


Cauchy Integral Formula

Closed contour integrals of non-analytic functions

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Taylor & Laurent Series Expansions

Derivation of both series using the Cauchy Integral Formula

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Residue Theorem

A very useful tool to have in your integration toolbox

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Derivation of the Planck Integral

Putting the results of the past 10 videos to use with Blackbody Radiation

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The following videos are made from footage taken from the parent videos above. They are meant to be short, sharp and to the point.

Video #

Video Tutorial Title


1 Solu­tion to the Planck Integral First half of PPV video 11 above


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