A function defined on an open set in the complex plane C is said to be differentiable at a point z = a if the limit of the difference quotient as it...
A function defined on an open set in the complex plane C is said to be differentiable at a point z = a if the limit of the difference quotient as it approaches a given point approaches a finite number. That is,
where L is a finite real number. This means that the derivative of the function at that point is equal to the limit of the difference quotient.
There are two main sufficient conditions for differentiability:
First condition: The function must be continuous at the point in question. This means that the limit of the function as it approaches the point in the direction of any real number should equal the value of the function at that point.
Second condition: The function must be analytic at the point in question. This means that the function must be defined and differentiable at all points in the domain of the function in a neighborhood of the point.
In other words, the first condition ensures that the function is not "too irregular" at the point, while the second condition ensures that the function has a well-defined derivative at that point.
Examples of functions that are differentiable include polynomials, exponentials, and trigonometric functions. Examples of functions that are not differentiable include polynomials with holes, exponential functions with negative exponents, and trigonometric functions with jumps or discontinuities at the endpoints of their domains.
The Cauchy-Riemann equations are a set of conditions that must be satisfied by a function in order to be differentiable at all points in its domain. These conditions ensure that the function has a well-defined derivative and that its derivative is continuous