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1058:{\displaystyle {\begin{aligned}&\int _{0}^{\infty }t^{z-1}e^{-t}dt\\&=\lim _{n\rightarrow \infty }\int _{0}^{n+1}t^{z-1}\left(1-{\frac {t}{n+1}}\right)^{n}dt\\&=\lim _{n\rightarrow \infty }{\frac {1}{z(z+1)\ldots (z+n-1)}}{\frac {n!}{(n+1)^{n}}}\int _{0}^{n+1}t^{z+n-1}\left(1-{\frac {t}{n+1}}\right)^{0}dt\\&=\lim _{n\rightarrow \infty }{\frac {n!}{z(z+1)\ldots (z+n-1)}}{\frac {1}{(n+1)^{n}}}{\frac {(n+1)^{z+n}}{z+n}}\\&=\lim _{n\rightarrow \infty }{\frac {n!}{z(z+1)\ldots (z+n)}}(n+1)^{z}\\&={\frac {1}{z}}\prod _{n=1}^{\infty }{\frac {(1+1/n)^{z}}{1+z/n}}\end{aligned}}}
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I've made it consistent. Since I had to choose one, I chose log, with a notation on first use that it is the natural logarithm. I think this is the more common style for mathematical analysis and analytic number theory, the main subfields of mathematics for this topic.
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What is t? This is crucial really, but I have never found a definition for it. Is it a constant? If not, how would you go about working it out on a scientific calculator with standard trig and logarithmic functions?
1553:{\displaystyle {\begin{aligned}\Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}dt\\\Gamma (x)=\int _{0}^{1}(-\ln(t))^{x-1}dt\\\Gamma (x)=\lim _{n\to \infty }n^{x-1}\prod \limits _{k=1}^{n}{\frac {1}{k+x-1}}\end{aligned}}}
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1976:. Obviously. That's what the "dt" part at the end of the integral denotes. If you don't know what a variable of integration is, this article may not be for you. β
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So when it is brought to the left of the equals sign as in the equation following "Laplace transform", it should have a negative sign.
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Knowledge. If you would like to participate, please visit the project page, where you can join
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I don't care much about which, but the article should be consistent. Do we want to use "
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If this observation is correct, I hope someone familiar with this subject can fix this.
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I believe that you are talking about two different expressions. Both of these are true:
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1835:{\displaystyle \lim _{n\to \infty }{\frac {1}{z}}\prod _{k=1}^{n}\left=\Gamma (z)}
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The reference for Euler's infinite product says that the limit goes to n!, not 1.
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The equation immediately following the words "Laplace transform" appears wrong.
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You mean the t in the first displayed equation and in the infobox? It's the
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Is there value in using this instead or additionally?:
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