What Is the Integral of the nth Derivative from Zero to Infinity?

[seanhbailey]

Homework Statement

In general, what is $$\int_{0}^{\infty} f^{(n)}(z) dn$$?

Homework Equations

The Attempt at a Solution

Is the answer as simple as taking the antiderivative of n?

 

[seanhbailey]

n is the nth derivative of f(z), and z is a constant. I want to integrate with respect to the variable n, the nth derivative. If I simply took the aniderivative of n, and if the values I wanted to evaluate the integral over were 1 and 0, I would obtain $$f^{(1/2)}(z)$$ which does not make any sense. What am I doing wrong?

 

[Mark44]

The whole thing seems screwy to me. Your limits of integration are 0 to infinity, but derivatives make sense only for integer values of n. E.g., the "one-halfth" derivative doesn't make any sense.

 

[seanhbailey]

Does the integral even have an antiderivative, forgetting about the limits?

 

[seanhbailey]

This is the problem I am encountering in the Euler- MacLaurin expansion in my proof. The proof is given in another one of my posts called Zeta Function Proof. If anyone would like to point me in the right direction, I would appreciate it very much.