Yes, I was using Weirstrass' theorem to show that an uncountable subset of the plane must have a limit point in it.mathwonk said:(as you know), even a countable convergent sequence of zeroes suffices to make a holomorphic function dead zero. the basic point is that an analytic entire function is determined by its power series coefficients, i.e. its derivatives at the center of the expansion, say at z = zero.
An identical zero function in the complex plane is a function that maps every complex number to zero, regardless of its input. It is a special case of a constant function, where the output is the same for all inputs.
An identical zero function is different from a zero function because it is defined to be zero for all inputs, while a zero function may have different outputs for different inputs.
Some properties of an identical zero function include being continuous, differentiable, and holomorphic (complex differentiable) everywhere in the complex plane. It also has a constant value of zero for all inputs.
The most common example of an identical zero function is the zero function itself, where the output is always zero for all inputs. Other examples include the function f(z) = 0, where z is a complex number, and the function g(z) = 0 + 0i, which is equivalent to the zero function.
The identical zero function plays an important role in mathematics and science as it is used as a starting point for many mathematical proofs and serves as a building block for more complex functions. It also has applications in physics, engineering, and other fields where complex numbers are used to model and solve problems.