There is a theorem that states that if f is analytic in a domain D, and the closed disc {z:|z−α|≤r} contained in D, and C denotes the disc’s boundary followed in the positive direction, then for every z in the disc we can write:

f(z)=12πi∫f(ζ)ζ−zdζMy question is:

What is the intuitive explanation of this formula?(For example, but not necessary, geometrically.)(Just to clarify – I know the proof of this theorem, I’m just trying to understand where does this exact formula come from.)

**Answer**

If you are looking for intuition then let us *assume* that we can expand f(ζ) into a power series around z: f(ζ)=∑n≥0cn(ζ−z)n. Note c0=f(z).

If you plug this into the integral and interchange the order of integration

and summation then that integral on the right side of the formula

becomes ∑n≥0∫cn(ζ−z)n−1dζ. Let us also assume that an integral along a contour doesn’t change if we deform the contour continuously through a region where the function is “nice”. So let us take as our path of integration a circle going once around the point z (counterclockwise). Then you are basically reduced to

showing that ∫(ζ−z)mdζ is 0 for m≥0 and is 2πi for m=−1.

These can be done by direct calculations using polar coordinates with ζ=z+eit. Now divide by 2πi and you have the formula. Of course this is a hand-wavy argument in places, but the question was not asking for a rigorous proof. Personally, this is how I first came to terms with understanding how Cauchy’s integral formula could be guessed.

**Attribution***Source : Link , Question Author : Pandora , Answer Author : KCd*