Integral representation of the Euler-Mascheroni constant

In summary, the conversation is about trying to prove a specific representation of Euler's constant, but the speaker is struggling to find a proof. They ask for help and mention that they have searched online but could not find the proof. The representation is shown as \gamma=-\int_{0}^{\infty}e^{-t}\log(t)dt and the speaker has tried integrating by parts and using power series, but neither have worked. They then mention finding an article on page 15 of a website, which provides a proof using the Weierstrass product of the Gamma function. The speaker expresses surprise that they did not come across this proof in their previous online search.
  • #1
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I am trying to prove a specific representation of Euler's constant, but I am not really getting anywhere. I hoped you could help me with this one, because I looked it up on the Internet and even though the relation itself is found in many webpages, its proof is in none. The relation is
[tex]\gamma=-\int_{0}^{\infty}e^{-t}\log(t)dt[/tex]
I tried integrating by parts and integrating term by term using power series, but none of them show the identity. Thanks for your help from now.
 
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  • #3
Thanks for the article. It comes up with a pretty good proof using the Weierstrass product of the Gamma function.

I sincerely wonder how I could not find that one with two hours of searching.
 

FAQ: Integral representation of the Euler-Mascheroni constant

What is the integral representation of the Euler-Mascheroni constant?

The integral representation of the Euler-Mascheroni constant (γ) is an expression that relates it to the natural logarithm function. It is defined as the limit of the difference between the harmonic series (the sum of the reciprocals of positive integers) and the natural logarithm function as the number of terms approaches infinity.

How is the integral representation of the Euler-Mascheroni constant derived?

The integral representation of the Euler-Mascheroni constant can be derived using techniques from calculus, specifically the integral test for convergence of infinite series. By comparing the harmonic series to the natural logarithm function, we can show that the limit of their difference is equal to the Euler-Mascheroni constant.

What is the significance of the integral representation of the Euler-Mascheroni constant?

The integral representation of the Euler-Mascheroni constant is significant because it provides a way to approximate the constant using a finite number of terms. This makes it useful in various mathematical and scientific calculations, such as in number theory and statistical analysis.

How is the Euler-Mascheroni constant related to other mathematical constants?

The Euler-Mascheroni constant is closely related to other important mathematical constants, such as π (pi) and e (Euler's number). For example, the integral representation of γ involves the natural logarithm function, which is defined in terms of e, and it also appears in various formulas involving π.

Are there other representations of the Euler-Mascheroni constant?

Yes, there are several other representations of the Euler-Mascheroni constant, such as infinite series, continued fractions, and integral expressions involving trigonometric functions. These representations are useful in different contexts and can provide alternative ways to approximate the constant.

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