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Taylor's Theorem

This thorem says that any analytic function in a circular domain has a power series expansion.

Taylor's Thorem

Let \(\Omega\) be an open connected set and let \(a\in \Omega\). Let \(f:\Omega→\mathbb{C}\) be analytic on \(\Omega\). Then there exists an \(r>0\) such that \(\(f(z)=\sum_{n=0}^\infty \dfrac{f^{(n)}(a)}{n!}(z−a)^n; \quad z ∈ D(a,r).\)\) In other words, the Taylor series of \(f\) at \(a\) converges uniformly and absolutely to \(f\) for all \(z∈D(a,r)\).

print("Hello World!")

sequenceDiagram
  autonumber
  Alice->>John: Hello John, how are you?
  loop Healthcheck
      John->>John: Fight against hypochondria
  end
  Note right of John: Rational thoughts!
  John-->>Alice: Great!
  John->>Bob: How about you?
  Bob-->>John: Jolly good!

Last modified on: 2023-01-04 23:20:05