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On my opinion, there are two very general reasons why analytic functions are important.

1. Solutions of many (almost all) important differential (and functional) equations are analytic. For example, all elementary and special functions arise in this way. Moreover, they are also usually analytic functions of parameters.

2. Harmonic analysis. Fourier (Laplace) transforms. This includes generating functions as a special case. They are in many cases analytic.

I think almost every occurence of analytic functions in mathematics and in real life can be traced to one of these two general sources.

Both these fundamental motivations apply not only to analytic functions of one variable but to analytic functions of several variables.

Historically, it seems that the first analytic functions of several variables that were studied were Abelian functions, which occur in the inversion problem of Abelian integrals (that is they occur as solutions of certain systems of differential equations). These differential equations appear both in pure mathematics (inversion of Abelian integrals) and in physics (integrable systems). They were studied long before the general theory of several complex variables was developed.

Then comes the second source (already in XX century): Wiener-Paley theorem and its multi-dimensional generalizations, Fourier analysis, which itself was developed as a method of solving PDE.

You specifically mention applications in algebraic geometry? Just open Griffiths Harris book! And for applications in "real analysis", look at least at the table of contents of Hormander's Linear differential operators!

On my opinion, there are two very general reasons why analytic functions are important.

1. Solutions of many (almost all) important differential (and functional) equations are analytic. For example, all elementary and special functions arise in this way. Moreover, they are also usually analytic functions of parameters.

2. Harmonic analysis. Fourier (Laplace) transforms. This includes generating functions as a special case. They are in many cases analytic.

I think almost every occurence of analytic functions in mathematics and in real life can be traced to one of these two general sources.

Both these fundamental motivations apply not only to analytic functions of one variable but to analytic functions of several variables.

Historically, it seems that the first analytic functions of several variables that were studied were Abelian functions, which occur in the inversion problem of Abelian integrals (that is they occur as solutions of certain systems of differential equations). These differential equations appear both in pure mathematics (inversion of Abelian integrals) and in physics (integrable systems). They were studied long before the general theory of several complex variables was developed.

Then comes the second source (already in XX century): Wiener-Paley theorem and its multi-dimensional generalizations, Fourier analysis, which itself was developed as a method of solving PDE.