Investigating Inconsistencies in Strogatz's Nonlinear Dynamics Book

[Dustinsfl]

Strogatz's Nonlinear and Dynamics book states that
$$
\langle\sin^{2n}\rangle = \frac{1\cdot 3\cdot 5\cdots (2n-1)}{2\cdot 4\cdot 6\cdots 2n}
$$
for $n\geq 1$.
However, $\langle\sin^6\rangle = \frac{5}{16}\neq\frac{15}{48}$.

What is the deal here?

 

[topsquark]

Strogatz's Nonlinear and Dynamics book states that
$$
\langle\sin^{2n}\rangle = \frac{1\cdot 3\cdot 5\cdots (2n-1)}{2\cdot 4\cdot 6\cdots 2n}
$$
for $n\geq 1$.
However, $\langle\sin^6\rangle = \frac{5}{16}\neq\frac{15}{48}$.

What is the deal here?

Ummm...
$$\frac{5}{16} = \frac{15}{48}$$

Or do we need numerator and denominator to be relatively prime? In that case they are not "equal"?

-Dan

 

[Dustinsfl]

Ummm...
$$\frac{5}{16} = \frac{15}{48}$$

Or do we need numerator and denominator to be relatively prime? In that case they are not "equal"?

-Dan

I apparently can't do math.

 

[Dustinsfl]

So I looked at
$$
\left\langle\left(\frac{e^{ix}-e^{-ix}}{2}\right)^6\right\rangle = -\frac{5}{16}
$$
The rest is zero due the inner product. So why am I getting a negative with this method when it should be a positive?

 

[Sudharaka]

So I looked at
$$
\left\langle\left(\frac{e^{ix}-e^{-ix}}{2}\right)^6\right\rangle = -\frac{5}{16}
$$
The rest is zero due the inner product. So why am I getting a negative with this method when it should be a positive?

Hi dwsmith, :)

Well I think you are missing the imaginary unit that should be in the denominator.

\[\sin{x}=\frac{e^{ix}-e^{-ix}}{2i}\]

Kind Regards,
Sudharaka.

 

[Dustinsfl]

Strogatz's Nonlinear and Dynamics book states that
$$
\langle\sin^{2n}\rangle = \frac{1\cdot 3\cdot 5\cdots (2n-1)}{2\cdot 4\cdot 6\cdots 2n}
$$
for $n\geq 1$.

How is this proved?

 

[chisigma]

Strogatz's Nonlinear and Dynamics book states that
$$
\langle\sin^{2n}\rangle = \frac{1\cdot 3\cdot 5\cdots (2n-1)}{2\cdot 4\cdot 6\cdots 2n}
$$

for $n\geq 1$...

How is this proved?...

First it is usefule to discuss a bit about what You mean as 'inner product'. According to...

Inner Product -- from Wolfram MathWorld

... in the space of real functions the 'inner product' of two functions f(*) and g(*) is defined as...

$\displaystyle \langle f(x) , g(x) \rangle = \int_{a}^{b} f(x)\ g(x)\ dx$ (1)

In the case of $f(x)=g(x)= \sin^{n} x$, $a=0$ and $b=\frac{\pi}{2}$ is...

$\displaystyle \langle f(x) , g(x) \rangle = \int_{0}^{\frac{\pi}{2}} \sin^{2 n} x\ dx = \frac{ 1\cdot 3\cdot 5\ ...\ (2n-1)}{2\cdot 4\cdot 6\ ...\ 2n}\ \frac{\pi}{2}$ (2)

You arrive at (2) using iteratively the integration by part...

$\displaystyle \int \sin^{m} x\ dx = - \frac{\sin^{m-1} x \cos x}{n} + \frac{m-1}{m}\ \int \sin^{m-1} x\ dx$ (3)

Kind regards

$\chi$ $\sigma$