Abstract
In two letters written as 1729 turned into 1730, the great Euler created what is today called the gamma function, Γ(n), defined today in textbooks by the integral
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Notes
- 1.
Newton didn’t actually say this (what is known as the second law of motion), but rather the much more profound ‘force is the rate of change of momentum.’ If the mass doesn’t change with time (as is the case in our problem here) then the above is okay, but if you want to study rocket physics (the mass of a rocket decreases as it burns fuel and ejects the combustion products) then you have to use what Newton really said.
- 2.
Notice that sign of dt is written with the ambiguous ±. I’m doing that because it’s not clear at this point (at least it isn’t to me!) which of the two possibilities coming from the square-root operation is the one to use. Physically, though, we know T > 0, and we’ll eventually use that condition to make our decision.
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Nahin, P.J. (2015). Gamma and Beta Function Integrals. In: Inside Interesting Integrals. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1277-3_4
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DOI: https://doi.org/10.1007/978-1-4939-1277-3_4
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