What Does it Mean for a Function to be Smooth on the Complex Plane?

In summary, the term "smooth" function refers to a function that satisfies certain regularity conditions, often denoted as C^2 or C^{infinity}. It may also be used in the context of holomorphic functions, but this is not a common usage.
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What is a "smooth" function?

If someone tells you to consider a "smooth" function on the complex plane, what does that mean, exactly? Does it mean that the partial derivatives of the real and imaginary parts exist and are continuous? Does it mean something else?
 
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Usually, "let f be a smooth function" is an expression people use to be "let f be a function satisfying certain regularity conditions".

Ex: for a smooth enough map f:R^2-->R, we have equality of the mixed partial derivative. Here, smooth means C^2.

Sometimes also, people equate smooth with C^{infinity}.

I've never seen smooth to mean holomorphic though...
 

1. What is a smooth function?

A smooth function is a mathematical function that is continuous and has derivatives of all orders. This means that the graph of the function has no abrupt changes or corners.

2. How is a smooth function different from a non-smooth function?

A smooth function is different from a non-smooth function in that it is continuous and has derivatives of all orders, while a non-smooth function may have abrupt changes, corners, or undefined derivatives at certain points.

3. What are some examples of smooth functions?

Some examples of smooth functions include polynomials, trigonometric functions, exponential functions, and logarithmic functions.

4. How are smooth functions used in science?

Smooth functions are used in science to model and describe various physical phenomena, such as the motion of objects, growth of populations, and changes in temperature or pressure. They are also used in data analysis and optimization problems.

5. Can a smooth function have a finite number of points where it is not smooth?

No, a smooth function must be smooth at every point in its domain. A finite number of points where the function is not smooth would make it a non-smooth function.

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