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I understand what analytic functions are and what differentiability of a complex function means but I have been reading "advanced engineering mathematics by kreyszig" and it says that the concept of analytic functions is motivated by the fact that differentiability at a point is of no practical use.

I can't seem to understand this , also the book states analyticity is important for "complex calculus" and I can't understand why just differentiability isn't enough .

An explanation using some examples would be appreciated , couldn't find an explanation in previous answers on this topic

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  • $\begingroup$ What is the defiition of analytic you are referring to? $\endgroup$
    – zhw.
    May 6, 2016 at 1:23
  • $\begingroup$ @zhw. The book I am referring to says that a function is analytic at a point if it has a derivative at every point in 'some' neighbourhood of that point $\endgroup$
    – Niyam Shah
    May 6, 2016 at 3:02

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Here are two important and basic theorems in complex analysis.

  1. Liouville's theorem: If $f\colon\mathbb{C}\to\mathbb{C}$ is entire (analytic everywhere) and bounded (so there exists $M\in\mathbb{R}$ with $|f(z)|\leq M$ for all $z\in\mathbb{C}$), then $f$ is constant.

  2. Cauchy's integral theorem: Let $U\subseteq\mathbb{C}$ be open and simply connected, let $f\colon U\to\mathbb{C}$ be analytic, and let $\gamma$ be a closed piecewise smooth contour in $U$. Then, $\int_{\gamma}f(z)\,\mathrm{d}z=0$.

Note that analyticity of $f$ is crucial to both of these theorems, and both of these theorems are important. Other examples of important theorems that rely on analyticity include Cauchy's integral formula, the Residue theorem and Rouché's theorem.

Why exactly is analyticity necessary? I don't want to get into technical details (you can find details easily enough in textbooks), so here's one way to look at it: there exist complex functions that are complex differentiable at precisely one point and nowhere else continuous.

That is, being complex differentiable at a point (and by extension, at lots of isolated points) does not ensure that the function is "well-behaved" enough for it to have "nice" properties like the above. We need a function to be complex differentiable everywhere in a particular region of the plane in order for it to be well-behaved enough for our usual purposes. Since complex differentiability in such a region provides analyticity in that region, this is a hand-wavy reason why analytic functions are the ones with which we work.

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  • $\begingroup$ That makes sense . Although can you recommend a good textbook , I know stack exchange isn't for book recommendations but I have two textbooks and none of them go into details over why analyticity is necessary sufficiently . $\endgroup$
    – Niyam Shah
    May 6, 2016 at 3:34
  • $\begingroup$ If you want a textbook for the details then the books by Priestley or by Stewart and Tall are good introductory texts. Another popular one is by Churchill and Brown - this one is known for its examples, you should take a look. $\endgroup$
    – Will R
    May 8, 2016 at 18:30

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