Wikipedia:Reference desk/Archives/Mathematics/2011 September 1

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

September 1

No (x) ?

Hi. Some math authors write functions f(x), g(x), etc solely as f or g without (x) in certain cases. For example, Spivak states IBP as: $\int {fg'}=fg-\int {f'g}$ When is this allowed, and why? danke. --Meumann — Preceding unsigned comment added by 24.92.85.35 (talk) 00:11, 1 September 2011 (UTC)

$f$ refers to the function itself whereas $f(x)$ refers to the value of $f$ evaluated at $x$Widener (talk) 00:14, 1 September 2011 (UTC)

For example, taking an antiderivative of a function is a transformation applied to the function, so using $f$ is preferable to $f(x)$ in this case. Widener (talk) 00:18, 1 September 2011 (UTC)

Of course, when $x$ is an independent variable, $f(x)$ is often interpreted as $x\mapsto f(x)$, which by extensionality is the same as $f$.--Antendren (talk) 01:41, 1 September 2011 (UTC)

Most of the time, this is just a shorthand notation used when it is clear from context where the function is evaluated. -- Meni Rosenfeld (talk) 07:08, 1 September 2011 (UTC)

Uniqueness and existence of the positive nth root

For positive $r,x$ and $n\in \mathbb {N}$ what is the proof that the polynomial $r^{n}=x$ has exactly one positive root? (as per nth root#Definition and notation) Widener (talk) 23:58, 31 August 2011 (UTC)

Suppose there are 2. Their quotient is a positive nth root of unity. Invrnc (talk) 00:33, 1 September 2011 (UTC)

$r^{n}$ is strictly increasing on $[0,\infty ]$, goes from 0 to $\infty$, and is continuous. Thus, it achieves every positive value (Intermediate value theorem) exactly once (being injective). -- Meni Rosenfeld (talk) 07:04, 1 September 2011 (UTC)

Analytic Functions

Considering the function $f\left(x\right)=\sum _{k=0}^{\infty }x^{k}{\sqrt {{\binom {k+t_{1}}{t_{1}}}{\binom {k+t_{1}}{t_{2}}}}}$ where$t_{1}>t_{2}$ and $t_{1},t_{2}$ are both natural numbers. Is it convergent? How do I tell? and does anyone know if there exists a more concise representation (without the summation)? — Preceding unsigned comment added by 192.76.7.237 (talk) 10:43, 1 September 2011 (UTC)

The expression ${\binom {k+a}{b}}$ where a, b are constant is polynomial in k. Thus, the coefficients in your power series have polynomial growth, so it converges for every $|x|<1$.

Mathematica is unable to find a closed-form expression for this, and I doubt one exists. -- Meni Rosenfeld (talk) 15:45, 1 September 2011 (UTC)

If you want to know how to see if a power series converges then have a look at our radius of convergence article. If you have a power series in a complex variable, say c₀ + c₁(z – a) + c₂(z – a)² + c₃(z – a)³ + …, then you use the coefficients c_k to calculate the radius of convergence. This is a (possibly infinite) positive real number ρ for which the power series converges for all | z – a | < ρ. If the radius of convergence is infinite then you have an entire function. — Fly by Night (talk) 16:06, 2 September 2011 (UTC)