Recent content by yasiru89

See http://mathrants.blogspot.com if anyone's interested in summation methods by which we may recover the analytic continuation of Riemann's zeta function.

If anyone's interested, I'm taking a look at divergent series and resummations at http://mathrants.blogspot.com

Mathtype might be the best way to go if what you need is the actual LaTeX code; I was going to suggest TeXnicCenter which I use with the MiTeX package and its quite good.

Throughout the logarithm is applied only to real quantities (like [itex]i^{i}[/tex]), so simply treating it as the real logarithm whenever the argument can at least be 'made' real would be justified (of course, for this we may require an alternate proof that [itex]i^{i}[/tex] is real, finding...

Perhaps the following might help, due to John Bernoulli which we might adapt for this case; Consider the area in the first quadrant of a unit circle centred about the origin. A = \int_{0}^{1} (1 - x^{2})^{1/2} dx With the change of variable [itex]u = ix[/tex] the integral is now, A =...

Oh yeah, Euler's result. There we are then.

The sum of reciprocals of divisors of a perfect number is always equal to 2 if the number itself is considered as a divisor. If, in your problem it is not, then we have that the sum of reciprocals of proper divisors is equal to 1 and the result is trivial. If it is the case that the standard...

You might try the Weierstrass product representation and have a go at \int_0^{+\infty} \frac{1}{\Pi(t)}dt , where [itex]\Pi(t) = \Gamma(t + 1)[/tex], but a nice answer probably won't result. You might try to get suitable Riemann sums at integer points on a general interval and try to bound...

Sounds a bit far-fetched, unless it was statistics or something, since real analysis is almost always a core component in undergraduate mathematics (though it may not be called that, for instance, in my university there's a Real and Complex analysis course offered for sophomores and then in...

Almost all universities offer something similar, there's not a great emphasis on number theory or foundations and discrete mathematics as I said before. Depending on that you can also avoid number theory at graduate level, sometimes even as a pure mathematician (though most have contact with...

Plenty do, don't they? At undergraduate level the main requirements (atleast for pure mathematics) seem to almost invariably be courses in algebra and analysis with appropriate electives. I'm only saying that number theory has its own (arguably supreme) beauty, just as the foundations of...

This is precisely why I chose to emphasise 'derivation', however, my criticism is not of the analytic extension as such, but of the practice to simply define, \exp{it} = \cos{t} + i\sin{t}. A better approach is to define the complex analogue as, \exp{z} = \sum_{n = 0}^{+\infty}...

Euler did use the series expansion, plugging in [itex]\theta := i\theta[/tex] in the defining series for the exponential, \exp{\theta} = \sum_{n = 0}^{+\infty} \frac{\theta^{n}}{n!} and seperating real and imaginary parts. It was a phenomenal achievement in the true Eulerian spirit. A bit...

An approximation with this particular function where the terms increase for real [itex]x[/tex], makes any such process largely meaningless. As has been said, there is no elementary anti-derivative (barring the infinite series version), for e^(-x^2) from 0 to infinity this is handled by...

Clearly the significance of number theory at the very foundations of mathematics is lost on some people. Look up Godel numbering and Godel's theorems of incompleteness. One of my personal favorite areas is analytic number theory, its almost a contradictory field, using analysis to deal with...