Successive derivative of analytic function

In summary, the given statement shows that for an analytic function f on a disk of radius R, if the sequence of its derivatives converges uniformly to f, then f can be expressed as a constant times the exponential function of z. This can be proved using Bach's differentiation formula or a simple limit argument.
  • #1
sbashrawi
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Homework Statement


Let f be analytic function on |z|<= R and lim fn(z)( the nth derivative) = f(z) uniformly in |z| <=R. Show that f(z) = c* exp(z)


Homework Equations





The Attempt at a Solution



I tried the power series expansion but I couldn't prove it
 
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  • #2
sbashrawi said:
Let f be analytic function on |z|<= R and lim fn(z)( the nth derivative) = f(z) uniformly in |z| <=R. Show that f(z) = c* exp(z)
Let {Gn(z)} be a sequence of analytic functions converging to G(z), an analytic function. If the convergence is uniform in |z| <=R, then Gn'(z) converges to G'(z). (This can be proved either by an elementary limit argument or by Bach's differentiation formula).
Here, Gn = fn -> f. As Gn' = fn'(z) = f_(n+1)(z), the same sequence converges to f' as well. Hence, f=f'. Taking h(z) =f(z) exp(-z), we find that h' =0 .QED.
 
  • #3
Thank you
 

What is the definition of a successive derivative of an analytic function?

The successive derivative of an analytic function is the derivative of the function's derivative. In other words, it represents the rate of change of the rate of change of the original function.

How is the successive derivative of an analytic function calculated?

The successive derivative of an analytic function can be calculated using the standard rules of differentiation, such as the power rule, product rule, and chain rule. These rules are applied to the previous derivative to obtain the next one.

What is the significance of successive derivatives in mathematics?

Successive derivatives are important in mathematics as they allow us to analyze the behavior of a function at a more detailed level. They can provide information about the concavity, inflection points, and extrema of a function.

Can the successive derivative of an analytic function be infinite?

No, the successive derivative of an analytic function is a finite value. This is because the function is assumed to be infinitely differentiable, meaning that it has derivatives of all orders.

How do successive derivatives relate to the Taylor series of a function?

The Taylor series of a function is a representation of the function as an infinite sum of its successive derivatives evaluated at a specific point. In other words, the successive derivatives determine the coefficients of the Taylor series, which allows us to approximate a function with a polynomial.

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