Hint: Writing the numerator of the fraction in the limit as $$p \left( e^{- \frac{y}{1+p}} + e^{- \frac{y}{1-p}} \right) + e^{- \frac{y}{1+p}} - e^{- \frac{y}{1-p}}$$ should help.

The case $1 = 0$ occurs when you are talking about the trivial ring $R = \{0\}$. It turns out $R$ is the trivial ring if and only if $1 = 0$, so stating $1 \neq 0$ is arguably the easiest way to say $... View answer 3 votes I would also add to Eric's answer that Tenenbaum's "Introduction to Analytic and Probabilistic Number" is a great resource if one is limited to the number of books you can carry. View answer Accepted answer 0 votes Write$P(x) = \displaystyle\sum_{n=0}^k a_n x^n$, and note$|P(x)| \leq k|a_k|x^k$for all$x \geq 1. Hint: \begin{align} |P(x)| \leq C_k x^k &\Longrightarrow \log P(x) \leq \log (C_k x^k) ... View answer 1 votes Q1. For the first question, since G is a group, each element a_i has an inverse a_i^{-1}. Note that inverses are unique. Let \{b_i\}_{i=1}^k denote the subset of elements of order 2 (I.e., ... View answer Accepted answer 31 votes Since x \mapsto \log x is a continuous function, and since continuous functions respect limits: \lim_{n \to \infty} f(g(n)) = f\left( \lim_{n \to \infty} g(n) \right), $$for continuous ... View answer 3 votes To graph these equations, I would just write out z = x + iy, and then use a standard graphing device, recognizing that we can identify \mathbb{C} with the plane \mathbb{R}^2. For instance, for ... View answer Accepted answer 8 votes Hint: Using induction: Show the base case n = 2. Show that if the statement is true for 2^n, then it is true for 2^{n+1}. Show that if the statement is true for n, then it is true for n - ... View answer Accepted answer 4 votes In practice, I prefer the (equivalent) following definition of upper Minkowski dimension:$$\overline{\dim}(E) = \limsup_{\delta \to 0} \frac{\log N(E,\delta)}{\log(1/\delta)}. $$N(E, \delta) is ... View answer 9 votes One of the most surprising results I have ever seen is the Universality Theorem of Voronin which states that any nonvanishing analytic function can be well -approximated by \zeta(s) somewhere in the ... View answer 2 votes As Jonas points out, a basis function is simply a function in the basis of a function space. Most likely, you are (whether it explicitly says so or not) working in the function space L^2[0,1] (or ... View answer Accepted answer 1 votes I hope I'm not misunderstanding the question, but here's what I think (I apologize ahead of time as it is late!) Write \eta(t) = \displaystyle\mathop{\Delta}_{t} \left[\frac{\phi(t) e^{-itx}}{it} \... View answer 9 votes The equal sign "=" should be used when you have two quantities which are equal. For example:$$x^2 + 3x - 4 = (x+4)(x-1).$$The "\Rightarrow" should be used when relating ... View answer 6 votes Hint: Write b^n = q_1^{n \alpha_1} \dots q_k^{n \alpha_k}, where \alpha_i > 0. If a^n | b^n, then a^n = q_1^{n \beta_1} \dots q_k^{n \beta_k}, where 0 \leq \beta_k \leq \alpha_k. View answer 2 votes HINT:$$\prod_{i=2}^n \left( 1 - \frac{1}{i^2}\right) = \prod_{i=2}^n \left(\frac{i^2 - 1}{i^2} \right)=\prod_{i=2}^n \frac{(i-1)(i+1)}{i^2} = \frac{\displaystyle\prod_{i=2}^n (i-1) \prod_{i=2}^n (i+1)... View answer 7 votes The (principal value) of the complex logarithm is defined as\log z = \ln |z| + i Arg(z)$. Therefore, $$\log(-1) = \ln|-1| + i Arg(-1) = 0 + i \pi.$$ and then, one simply gets$$-i \log(-1) = -... View answer Accepted answer 1 votes Jason. You should not be solving for$y$, you should be solving for$\frac{dy}{dx}$. There was an error in your calculation, however, when you applied the chain rule to find$\frac{d}{dx} (\sin (\pi ...

To supplement Derek's answer, the Riemann zeta function $\zeta(s) = \sum_{n} n^{-s}$ was originally only considered for $s = \sigma + i t$, where $\sigma > 1$. Note that when $\sigma > 1$, the ...

If $g(x)$ and $f(x)$ tends to $\infty$, then there is a value $x_0$ such that for $x > x_0$, $g(x)$ and $f(x)$ are strictly positive. Therefore, if $-C \leq f(x) - g(x) \leq C$, then for $x > ... View answer Accepted answer 4 votes Add xy to both sides (To get$y'' = xy$), and start taking derivatives. A pattern emerges. Edit: You should get:$y^{(3)} = y + xy'y^{(4)} = 2y' + xy''y^{(5)} = 3y'' ...

This may or may not be what you're looking for, but I thought the idea was fun, so I'll share: This will be a non-symmetric metric on $\mathbb{Z}^+$: Suppose that one is only allowed to move left ...

This is an interesting take: http://blogs.ams.org/mathgradblog/2010/11/17/the-minimalist-approach-to-an-advisor/ ${}{}{}{}{}$

The $\$27$includes the$\$25$ they paid for food, plus the $\$2$to the waiter. They pocketed the other three dollars. View answer 6 votes The Kakeya conjecture. It states that a set in$\mathbb{R}^d$containing a line in every direction has Hausdorff dimension$d$. It is solved for$d = 2$and open for$d \geq 3\$. The finite field ...