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

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Homework Help Overview

The discussion revolves around the integral of the nth derivative of a function from zero to infinity, specifically questioning the validity and interpretation of integrating with respect to the derivative order n.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of integrating the nth derivative and question the meaning of non-integer derivatives. There is a discussion about the validity of taking the antiderivative of n and the resulting expressions.

Discussion Status

The conversation is ongoing, with participants expressing confusion about the limits of integration and the nature of derivatives. Some have raised concerns about the feasibility of the integral and the interpretation of fractional derivatives, indicating a lack of consensus on the approach.

Contextual Notes

There are discussions about the constraints of integrating derivatives, particularly the assumption that derivatives are only defined for integer values of n. The context of the Euler-Maclaurin expansion is also mentioned, suggesting additional complexity in the problem.

seanhbailey
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Homework Statement



In general, what is [tex]\int_{0}^{\infty} f^{(n)}(z) dn[/tex]?

Homework Equations





The Attempt at a Solution


Is the answer as simple as taking the antiderivative of n?
 
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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 [tex]f^{(1/2)}(z)[/tex] which does not make any sense. What am I doing wrong?
 
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.
 
Does the integral even have an antiderivative, forgetting about the limits?
 
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.
 

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