Steiner

Can he do anything?

For a> 0,  determine the function F defined by

F(a) = Integral (0, oo) ln(x)/(x^2 + a2)^2 dx

The answer is 

F(a) = pi (ln(a) - 1)/(4a^3)

I usually write technical reports in English (I'm not a native) and often use the word aproximately. I was told this is a very formal and technical word that should be avoided in normal speech. 

What's a good word to use instead? Maybe about? For example, instead of saying something costs aproximately 10 dollars should I say it costs about 10 dollars?

Thank you.

Suppose you are an employee of a bank. Then, should you say

I work in, for or at a bank?

Thank you.

Let f and g be entire functions such that lim |z| ----> oo f(z)/g(z) = 1. Show that f and g have finitely many zeroes on C (the set of complex numbers) and that f and g have the same number of zeroes.

Thank you.

In equation, tion is pronounced like sion in version.Very different from it's pronunciation in most words ending in tion, such as nation, organization, administration and so on.

Why is that? Or is that one of those mysteries of English that no one can explain?

Thank you.

I couldn't prove this so far. I guess a possible proof is based on the Open Map Theorem. Can anyone help?

Suppose f is holomorphic in the open disk D(0, 1). Let u and v be the real and imaginary parts of f and suppose |u| + |v| = 1 all over D(0, 1). Show that f is constant.

Thank you.

Thank you.

Uma das músicas era Slave to Love.

Eu gostaria de ter as faixas dele.

Muito obrigado.

I think this is interesting.

Let P_ n be the polynomial defined on the complex plane by

P_n(z) = (z^n) (z - 2) - 1, n positive integer,

and let c be the boundary of the open disk D(0, 1). Show that

1) I_n = ∫_c dz/(P_n(z)) exists for all n

2) Out of the n + 1 zeroes of P_n (counting multiplicities), there's a particular one, z_n, such that you can express I_n, in closed form, as a function of n and z_n.

3) lim n → ∞ I_n = 0

Thank you.

I think this is interesting.

Let P_ n be the polynomial defined on the complex plane by

P_n(z) = (z^n) (z - 2) - 1, n positive integer,

and let c be the boundary of the open disk D(0, 1). Show that

1) I_n = ∫_c dz/(P_n(z)) exists for all n

2) Out of the n + 1 zeroes of P_n (counting multiplicities), there's a particular one, z_n, such that you can express I_n, in closed form, as a function of n and z_n.

3) lim n → ∞ I_n = 0

Thank you.

I can't see what it has to do with the tail of a cock.

Thank you.

I think this is interesting.

Let P_ n be the polynomial defined the complex plane by

P_n(z) = (z^n) (z - 2) - 1, n positive integer,

and let c be the boundary of the open disk D(0, 1). Show that

1) I_n = ∫_c dz/(P_n(z)) exists for all n

2) Out of the n + 1 zeroes of P_n (counting multiplicities), there's a particular one, z_n, such that you can express I_n, in closed form, as a function of n and z_n.

3) lim n → ∞ I_n = 0

Thank you.

I couldn't prove this, can anyone help?

Suppose the nonconstant f is holomorphic in a convex open set V and Re(f(z)) >= 0 for all z in V. Show that f is one-to-one.

Thank you.

Uma das músicas era Slave to Love.

Eu gostaria de ter as faixas dele.

Muito obrigado.

Não tenho acompanhado esta novela, mas ontem assisti à cena da abertura do Mar Vermelho. Com a tecnologia digital de hoje, esperava algo realmente espetacular. Mas me decepcionei. Acho que a cena que vi em 1960, no filme Os Dez Mandamentos, gravado em 1956, a cena foi mais bonita. Confiram abaixo.

Concordam comigo?

http://youtu.be/Jo0JMs-evQU

Suppose the improper Riemann integral of the real valued f over (a, b], a and b in R, is finite. Suppose, in addition, that f is bounded below by some real w and Let (P_n) be a sequence of partitions of [a, b] such that ||P_n|| --> 0. Then, is it true that

L(P_n, [a, b], f) --> Int (a, b] f(x) dx (improper integral) ?

I think it is, but would like someone to confirm this (or not, if I'm mistaken). I gave a proof based on the Dominated Convergence Theorem.

If we suppose f is continuous, the answer is indeed yes.

It's not supposed that f is bounded above on (a, b].

Thank you.

I think this is interesting. I invite you to answer. This was discussed here about 2 years ago.

Suppose there is a function f from R to R such that, for every real x, f(f(x)) = ax^2 + bx + c, a ≠ 0, b and c real coefficients. Show that

(b +1) (b - 3) ≤ 4ac

Happy new year 2015!

I have a small wound in my back. Very small, something like the bite of a mosquito. I've had it for more than 3 months. It itches and bleeds a bit if I scratch it with my nails.

Just like the bite of a mosquito. Maybe that's what it really is. The problem is it never heals completely. When I think it's gone, it starts itching again.

Can this be something to worry about? Should I see a doctor?

Thank you.

It's well known that if f from R to R is continuous, periodic and non constant, then f has a fundamental period. But I was told, without proof, that if f is periodic, non constant and continuous at a single element of R, then f has a fundamental period. That is, if f is periodic and non constant, then a sufficient condition for f to have a fundamental period is that f is continuous at a single real.

I tried but couldn 't prove this. Is it actually true?

Can anyone help?

Thank you.