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

{| width = "100%"

|- ! colspan="3" align="center" | Mathematics desk |- ! width="20%" align="left" | < August 31 ! width="25%" align="center"|<< Aug | September | Oct >> ! width="20%" align="right" |Current desk > |}

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded 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: $\int f{g}^{\prime }=fg-\int f^{\prime }g$ 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 $[0,\mathrm{\infty }]$, goes from 0 to $\mathrm{\infty }$, 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 $f\left(x\right)=\sum _{k=0}^{\mathrm{\infty }}{x}^k\sqrt{(\genfrac{}{}{0ex}{}{k+t_1}{t_1})(\genfrac{}{}{0ex}{}{k+t_1}{t_2})}$ 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 $(\genfrac{}{}{0ex}{}{k+a}{b})$ 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 Template:Nowrap, 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 Template:Nowrap. If the radius of convergence is infinite then you have an entire function. — Fly by Night (talk) 16:06, 2 September 2011 (UTC)