The M1M1 treasure hunt, week 9

Hermione, as usual, arrived early for the lecture. Somewhat to her surprise, Ron was already there. "He's late!" muttered Ron.
"It's already Week 10 and he hasn't yet released the Week 9 clues."
"He's probably trying to think of a way of making it fair for everyone. You've got all the passwords? Then all we can do is wait. Let me show you some interesting results about the asymptotics of integrals."
"Asym-what?"
"Asymptotics, like asymptotes. It means how things behave when something gets very large, a bit like limits. For example, take the integral of f(x)cos(nx), between 0 and 2π, say. As n → ∞ do you know what the answer is?"
"It depends on f(x) of course."
"No Ron, it doesn't. As n gets really big, the cosine oscillates so quickly between positive and negative values that they have to cancel. Provided f(x) is integrable, the limit has to be zero. This result was proved by

." ?

"Well, so what? Suppose f(x) is integrable. Then the integral of f(x)/n also tends to zero as n → ∞."
"How true!" smiled Hermione. "We'll call that Weasley's lemma. Then there's this nice result. Using a reduction formula, we can show that the factorial n! is the integral over (0,∞) of g(x), where g(x) = xⁿexp(-x) = exp(n log(x) -x). Now g(x) has a maximum when x=n. We write x=n+t and let n → ∞. We expand in t/n, do the t-integral and we find

n! = (2π n)^1/2(n/e)ⁿ.

Isn't that amazing? That formula was derived by

?

"What rubbish! The RHS isn't even a whole number," moaned Ron, who was somewhat lost.
"I mean asymptotically equal. If you like, it's a very good approximation when n is large."
Hermione sighed. "It's all so beautiful! I think while we were defeating you-know-who I neglected my Muggle side. Did you know that physical laws, which govern everything we do, tend to reflect some deep mathematical symmetry? For example, the law of momentum conservation, which we'll do next term in M1A1, is a consequence of the translational invariance of Euclidian space?"
Ron saw no point in pretence. "Hermione, I haven't the faintest notion what you're going on about. You're the cleverest girl I know by far. And the prettiest."
Hermione blushed. "Well the woman who invented that theorem was pretty clever too. Her name was

."?




Go to Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Week 9