Can You Integrate (ln ln x)^n Using a Taylor Series?

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Discussion Overview

The discussion revolves around the integration of the function (ln ln x)^n for any integer n, exploring methods for evaluating the integral, particularly through numerical approaches and Taylor series expansion.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant questions how to integrate (ln ln x)^n for any n.
  • Another participant suggests that since Mathematica cannot find a formula for n=2, numerical methods may be the only option.
  • A follow-up asks for clarification on what "numerically" entails, specifically whether it refers to evaluating a definite integral.
  • One participant asserts that finding a general closed-form expression for the indefinite integral is unlikely, proposing that definite integrals should be approached numerically.
  • Another participant claims that there exists an explicit evaluation for the integral of -log[-log[x]], which relates to the Euler constant, citing differentiation of the gamma function.
  • A participant reiterates the difficulty in expressing the integral in terms of elementary functions and questions whether (ln ln x)^n can be represented as a Taylor polynomial for integration purposes.

Areas of Agreement / Disagreement

Participants generally agree that finding a closed-form expression for the indefinite integral is challenging, with some advocating for numerical methods. However, there is no consensus on the feasibility of using Taylor series for integration.

Contextual Notes

Participants express uncertainty regarding the Taylor series for ln ln x and the implications of representing the function as a polynomial for integration.

flouran
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How do you integrate (ln ln x)^n for any n?
 
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Well since mathematica isn't able to find a formula for n=2, I'm going to say "numerically".
 
nicksauce said:
Well since mathematica isn't able to find a formula for n=2, I'm going to say "numerically".

What do you mean exactly by "numerically"? Do you mean that i should evaluate it as a definite integral?
 
Well I just mean if you want to find a general closed for expression for the indefinite integral, you are out of luck. Therefore the only way I can conceive of doing an integral with this expression would be to do a definite integral numerically.
 
there is one place where the integral can be evaluated explicitly that I know of.
\int-log[-log[x]]dx=Euler Constant (.577...)
This follows from differentionation the gamma function in its product and integral forms and making a change of variables.
 
nicksauce said:
Well I just mean if you want to find a general closed for expression for the indefinite integral, you are out of luck. Therefore the only way I can conceive of doing an integral with this expression would be to do a definite integral numerically.
Although this method would be painful, since I cannot express this integral in terms of elementary functions, could I represent (ln ln x)^(n) as a Taylor polynomial (what is the Taylor series for ln ln x, anyways?) and then integrate that and leave it as a Taylor Series?
 

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