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Wolfram Alpha » Explore anything with the first computational knowledge engine. The simplest examples of such functions are the common real functions that can be defined by a power series.
Just use the same power series, while allowing the function argument to be complex. Differentiability as a complex function is defined in the usual way as a limit at a point :. This looks just like the usual "real" definition, except that the absolute values are taken on complex numbers.
The absolute value of a complex number is just its modulus, that is, its distance from the origin. It is a real number. The open image theorem. We understand pretty well the connection between the position of zeroes of an analytic function and the structure of the function.
In short, it is possible to factor out the zeroes as we can do for polynomials. Now I would like to get a bit more in details into the examples so that we can understand in which fields analytic functions occur naturally. There are even more occurrences but I will stick to these three because they are the easiest to talk about.
How easy it is to meet analytic functions in math theories is probably partly explained by the various view points we can have on analytic functions: as power series, as derivable functions see below or as Cauchy integrals.
Just like mathematicians need to understand the asymptotic behaviour of real functions, they need to do so for complex functions. The main theorem here is. What a strong result! Nothing near this is true for real functions!
Stepping further in this direction will lead us to theorems like Dirichlet's theorem for subharmonic functions. Algebraic geometers study geometric objects that can be described by polynomial equations — usually with a lot of dimensions and variables. Now, when we learn group theory and differential calculus, we discover that polynomials are not always enough to study our problems and that it is very nice to have an emergency class of easy functions, with polynomials in it, the exponential and the logarithm.
Analytic functions are here a good fit. The first thorough study of numeric series was made by Euler and he made the following fascinating observation:. This is a very exciting observation because the left hand side is easily seen to be an analytic function and the right hand side tells something about the set of prime numbers, which is one of the primary study object in number theory!
So, in particular, you can do the whole real analysis in the realm of holomorphic functions, being now able to use powerful complex analysis techniques also to study pretty bad functions.
This seems at least formally like an interesting property to have. This is useful if you are in the s and need to compile a logarithm table. As Chappers says, the analytic property of a function is very useful on those defined on the complex plane, and it turns out that all the usual functions are analytic.
Those functions have very interesting properties, such as complex-derivative, zero integral on closed paths, and the residue formula. You can also use a lot of real analysis results Leibniz' formula, the chain rule in the study of analytic functions.
For instance, without complex-analytic related tools, it would be impossible to prove major theorems like the Prime Number Theorem. As a matter of fact, it is quite astounishing that properties of complex function need to be used to prove a result about arithmetics.
Taking a more "applied" approach to this question, I would say that analytic functions are so important because they come up in practical problems. Newton's laws of motion are differential equations, and analytic functions play well as solutions to differential equations. There are tons of other problems in physics and other sciences that use differential equations and benefit from analytic functions.
Analytic functions are well-behaved and easy to work with. Studying an analytic function is typically much easier than studying a non-analytic one. Also, I would say even just approximation by itself is important. Of course, analytic functions are not the solution to every problem.
An example are non-analytic smooth functions that can be constructed with Fourier series: a tool commonly used in signal processing and more Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why is the notion of analytic function so important? Ask Question. Asked 3 years, 5 months ago. Active 3 years, 4 months ago. Viewed 7k times.
Rodrigo de Azevedo Code Complete Code Complete 6 6 silver badges 12 12 bronze badges. In what context do they say it's important? Basically why are you under the impression that its of agreed upon "importance"? So the question is, why do so many authors believe they are important?
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