An analytic function is a mathematical function that can be expressed as a power series, meaning it can be represented by an infinite sum of terms with increasing powers of the independent variable. In other words, it is a function that can be differentiated and integrated infinitely many times over its domain.
A non-analytic function is a function that cannot be expressed as a power series. This means that it cannot be differentiated or integrated infinitely many times, and therefore does not have a well-defined derivative or integral at certain points in its domain.
To find the points where a function is not analytic, you can use several methods such as taking the derivative of the function and looking for points where it is not defined, or using the Cauchy-Riemann equations to check for differentiability at a certain point. You can also use graphical methods to identify points of non-analyticity.
Some common examples of non-analytic functions include functions with logarithmic or absolute value terms, functions with discontinuities or corners, and functions with points of singularity such as poles or branch points. Trigonometric and exponential functions can also be non-analytic at certain points.
Identifying points where a function is not analytic is crucial in many areas of mathematics and science. It can help determine the behavior and properties of a function, and can be used in applications such as optimization, numerical analysis, and solution of differential equations. It also allows us to better understand the structure and limitations of mathematical functions.