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    Equality of definite integrals, relation between integrands

    Suppose we are given two functions: f:\mathbb R \times \mathbb C \rightarrow\mathbb C g:\mathbb R \times \mathbb C \rightarrow\mathbb C and the equation relating the Stieltjes Integrals \int_a^\infty f(x,z)d\sigma(x)=\int_a^\infty g(x,z)d\rho(x) where a is some real number, the...
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    Gamma Function and the Euler-Mascheroni Constant

    The so called product integral was developed in 1887 by Volterra according to And here is the page for multiplicative calculus, just in case anyone is interested:
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    Gamma Function and the Euler-Mascheroni Constant

    Just in case anyone wanted to know, I found a lot about it. It's called Bigeometric (or Multiplicative) Calculus.
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    Gamma Function and the Euler-Mascheroni Constant

    Yea I think so too. This is a little bit off topic, but, it was also my reason for looking at the properties of the gamma function. Have you seen any theory on extending a product over a continuous interval, as is done with the sum to create the integral? I have tried to develop an approach...
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    Gamma Function and the Euler-Mascheroni Constant

    Actually, it follows immediately if you change the LHS of the last line to the equivalent -(log(gamma(1+s))-log(gamma(1))/s since log(gamma(1))=log(1)=0 and the digamma function ψ is defined to be the logarithmic derivative of the gamma function, so that under the limit as s ->0 this is...
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    Gamma Function and the Euler-Mascheroni Constant

    I didn't even think to look at digamma, but it seems that as usual, you are indeed correct. It follows straight from a series representation for it. Basically, the last line in my analysis is almost exactly this ψ(x+1)=-γ+∑(1/k-1/(x+k))
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    Gamma Function and the Euler-Mascheroni Constant

    I was taking a break from studying from my real analysis, electrodynamics, and nuclear physics exams this week, and, being a math-phile, I decided to play around with the gamma-function for some reason. Anyway, I used the common product expansion of the multiplicative inverse, and I arrived at a...
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    How to derive beta function as pochhammer contour integral?

    Awesome, thanks. I haven't used mathematica in a few years. Actually, last time I used mathematica I don't even think I knew what a contour was. I have been using python lately to do everything computationally/graphically. It would be awesome though to not have to literally write my own program...
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    The space of solutions of the classical wave equation

    No, he's right. Every solution is of that functional form. Every solution to the wave equation has a forward traveling wave and a backward traveling wave. However, the context of your conclusion solution is what is setting you off. First look at sturm-liouville theory, then learn some real and...
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    Baby Rudin Theorem 1.11

    You should definitely look at Real Mathematical Analysis by Charles Chapman Pugh. I find it much easier to read than Rudin. I honestly don't know why Baby Rudin is preferred by so many, I feel like Baby is sort of outdated and lacks organization of thoughts in some of his proofs. Your...
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    How to think of uniform continuity intuitively?

    I really don't know if there is a good intuitive approach to uniform continuity. I'll try... The difference between the two is that with regular continuity, the ball of radius delta about some point c depends on how far you are away from c in the domain (delta depends on x and and epsilon). In...
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    Interchaning Limits and Inner Products

    Exchanging limits and anything else, i.e. derivatives, sums, integrals depends on whether or not a sequence of functions is uniformly convergent. Since in Hilbert space, the inner product is either a sum for discrete or an integral for continuous cases, such a result is dependent on whether or...
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    How to derive beta function as pochhammer contour integral?

    There are a vast number of beautiful results in complex analysis. This is indeed one of them. May I ask what plotting utility you used?