Complex Analysis

As recently as 2022, Takuro Mochizuki won the Breakthrough prize in mathematics for his work at the intersection of complex analysis and differential geometry.

Many of the most important and significant theorems across all disciplines of mathematics belong to complex analysis in a visceral and real way:  the fundamental theorem of algebra, the theory of exact differential equations and potentials, and the residue theorem.  The most important underlying results here will be the Cauchy-Riemann equations (right) and the Cauchy integral formula (left) given below respectively:


$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ and $2\pi if(z_0)=\oint\frac{f(z)}{z-z_0}dz$.

The first third of this book will be given over to unpacking the notation and language of this pair of equations as well as proving preliminary results and building fundamentals in complex analysis.  The next third will be devoted to more specialized results and the final third will look particularly at common applications across many disciplines.  I have taken considerable inspiration from the complex analysis textbooks of John Conway, Jerrold Marsden, and of Brown and Churchill as well.  

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One thing I would like to particularly emphasize is that it is the multiplicative structure of the complex plane, which we will henceforth refer to as ℂ that yields these interesting and deep properties. Below you seen an example of this with the particular function f(z) = z .  Blue lines are mapped to blue lines and orange lines are mapped to orange lines.  We see the action of f(z) on a complex grid and this is one way of representing complex functions.