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...
The so called product integral was developed in 1887 by Volterra according to http://en.wikipedia.org/wiki/Product_integral
And here is the page for multiplicative calculus, just in case anyone is interested:
http://en.wikipedia.org/wiki/Multiplicative_calculus
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...
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...
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))
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...
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...
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...
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 I honestly don't know why Baby Rudin is preferred by so many, I feel like Baby is sort of outdated and lacks organization of thoughts in some of his proofs.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...
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...
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...