Here is a really quick derivation of the Cauchy-Riemann equations of complex analysis.
Consider a function of a complex variable, , where , such that:
where and are real-valued functions.
An analytic function is one that is expressible as a power series in .
We formally differentiate this equation as follows. First, differentiating with respect to , we obtain
Differentiating with respect to , we obtain
Multiplying the latter equation by and equating to the first result, we obtain
Comparing imaginary and real parts of these equations, we obtain
which are the famous Cauchy-Riemann equations.
3 replies on “A Really Quick Derivation of The Cauchy-Riemann Equations”
I have seen your paper on arXiv in which you had tried to solve the Shwarzschild black hole in the context if Schrodinger equation but left unfinished. I am curious to know if you have solved it or not.
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Thaank you for writing this