Complex Derivatives, Wirtinger View and the Chain Rule

Two days ago in Julia Lab,  Jarrett,  Spencer,  Alan and I discussed the best ways of expressing derivatives for automatic differentiation in complex-valued programs. Having inspired from this discussion, I want to share my understanding of the subject and eventually present a chain rule for complex derivatives.

Derivative

\mathbb{R}ealistic view: derivative is a real number that tells you how fast a value changes with respect to a variable.

D = \frac{dy}{dx} Continue reading

Gaussian Integral and A Pi Approximator

I remember how I was happy when I find the integration of the Gaussian function in my high school years. It was an instantaneous spark which make me realize that increasing the integration dimension may bring a solution. Moreover, one can find a series expansion for Pi number after evaluating the integral.

Gaussian Integral

What is the Gaussian integral: Finding the area under the Gaussian function. I = \int_{-\infty}^{+\infty}\, e^{-\alpha x^2}\,dx

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