Is the digamma function close to 0 for large arguments?

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In summary, the conversation discusses the limit of a term with a digamma function on the right-hand side of an equation, with the goal of showing that the term should be close to 0. The professor asks for justification using the integral form of the digamma function, but the person researching online finds that it either slowly diverges or converges to a number larger than 0. The integral form of the digamma function is discussed as a possible explanation.
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Judyxcheng
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1. The problem statement, all variables and given/known
I have taken the limit of both sides of an equation for x going toward infinity. There is a digamma (psi(x)) function on the RHS, and the limit of the term is supposed to be (at least close to) 0. Thus, the term can cancel out.

My professor said that indeed, the digamma function is supposed to be around 0 for large arguments and he wants me to justify that with the digamma function's integral form. However, when I research digamma online, it appears to slowly diverge or converge to a number larger than 0. Can someone clarify this?

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Integral form of digamma function, found here where it says "The digamma function satisfies": http://mathworld.wolfram.com/DigammaFunction.html
 
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  • #2
It does not go to zero, but it grows only as fast as a logarithm, you can see that based on the recurrence relation.
 

FAQ: Is the digamma function close to 0 for large arguments?

1. What is the digamma function and why is it important?

The digamma function, denoted by ψ(x), is a special function in mathematics that is defined as the logarithmic derivative of the gamma function. It is important in many areas of mathematics, including number theory, combinatorics, and physics.

2. How does the digamma function behave for small arguments?

For small arguments, the digamma function approaches negative infinity. Specifically, ψ(x) ~ -1/x as x approaches 0.

3. Is the digamma function close to 0 for large arguments?

No, the digamma function does not approach 0 for large arguments. In fact, it has a logarithmic growth rate, meaning it increases slowly as x gets larger.

4. Are there any special values for the digamma function?

Yes, the digamma function has some well-known special values, such as ψ(1) = -γ (where γ is the Euler-Mascheroni constant) and ψ(1/2) = -π - 2ln(2).

5. How is the digamma function used in practical applications?

The digamma function is used in various areas of mathematics, including statistics and differential equations. It also has applications in physics, particularly in the study of quantum mechanics and the Riemann zeta function.

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