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when is function analytic ?

**Understanding Analytic Functions: A Comprehensive Guide**

**Introduction to Analytic Functions**

In mathematics, the concept of analytic functions plays a crucial role, particularly in both real and complex analysis. An analytic function is one that can be locally represented by a convergent power series. This property distinguishes analytic functions from other functions, offering unique characteristics and applications.

**Definition of Analytic Functions**

An analytic function is defined as a function that is locally given by a convergent power series. For a function to be analytic at a point, the radius of convergence of its Taylor series must be positive, ensuring that the function equals its power series expansion in some neighborhood around each point.

**Real vs. Complex Analytic Functions**

– **Real Analytic Functions**: These functions are smooth (infinitely differentiable) and can be expressed as a convergent power series in some neighborhood of every point in their domain. However, not all smooth functions are analytic; for example, the bump function is smooth but not analytic.

– **Complex Analytic Functions**: In complex analysis, a function is analytic if it is complex differentiable in a neighborhood of every point in its domain. This is equivalent to being holomorphic. Complex analytic functions have additional properties, such as being conformal (angle-preserving) and satisfying the Cauchy-Riemann equations.

**Examples of Analytic Functions**

– **Real Analytic Functions**: The exponential function ( e^x ) and trigonometric functions like ( sin(x) ) and ( cos(x) ) are examples of real analytic functions. Their Taylor series expansions converge to the function in their domains.

– **Complex Analytic Functions**: Functions like ( e^z ), ( sin(z) ), and ( cos(z) ) are complex analytic. The function ( 1/z ) is analytic everywhere except at ( z = 0 ), where it has a singularity.

**Importance of Analytic Functions**

Analytic functions are significant due to their unique properties:

1. **Identity Theorem**: If two analytic functions agree on a set with a limit point in their domain, they agree everywhere in that domain. This theorem underscores the rigidity of analytic functions.

2. **Applications**: Analytic functions are pivotal in solving differential equations, conformal mapping, and various areas of physics and engineering, where their properties facilitate deeper analysis and solutions.

**Conclusion**

Analytic functions, whether real or complex, are fundamental in mathematics due to their ability to be expressed as convergent power series. Their properties, such as the identity theorem and applications in diverse fields, highlight their importance. Understanding the distinction between real and complex analytic functions, along with their examples and properties, provides a deeper insight into their role in mathematical analysis and beyond.

This guide aims to demystify the concept of analytic functions, offering clarity on their definitions, examples, and significance in the broader mathematical landscape.