I first noticed that three Youtube videos [1] [2] [3] give the same definitions of the Jacobi elliptic functions sn, cn, dn (see below), which are different from Wikipedia's. Then I noticed the links in this post (blog, published article, course notes) have the same (incorrect, I think) definitions as seen on Youtube. Two of these are by the same person, who is also the lecturer in one of the Youtube videos. I'm guessing the others are repeating his idea.
Here's their proposed definition for sn.
sn$(u,k)$ is the $y$ coordinate of the point $P_u$ on the ellipse $(x/a)^2+y^2=1$, where $k^2=1-1/a^2$, and the point $P_u$ makes an angle of $\phi$ with the $x$ axis such that $u=F(\phi,k)=\int_0^\phi \frac1{\sqrt{1-k^2\sin^2\theta}}d\theta$.
But this can't agree with the more common definition $\operatorname{sn}(u,k)=\sin\phi=\sin\operatorname{am}(u,k)$, which appears on Wikipedia and everywhere else I've looked, including Math.SE. Indeed, with $y$ as above, we have $y=r\sin\phi>\sin\phi$, whenever $a>1$.
They define cn$(u,k)=x_P/a$, where $x_P$ is the $x$-coordinate of $P$ as defined above, and dn$(u,k)=r/a$.
- Am I correct that these alternative definitions of sn/cn/dn are wrong?
- How wrong are they (e.g., are they just off by a constant factor), and can the definitions be easily salvaged?
With regard to (2), note that all these sources seem to derive a lot of correct formulas, which demands some explanation if the starting point is indeed wrong.
$\DeclareMathOperator{\sn}{sn}\DeclareMathOperator{\cn}{cn}\DeclareMathOperator{\am}{am}$ I looked a little at the first video (William Schwalm's lecture). There he uses an angle $\theta$ which is such that the point $P = (x,y)$ on the ellipse $(x/a)^2 + y^2 = 1$ (with $a>1$, and hence eccentricity $k = \sqrt{a^2-1}/a$) satisfies $$ (x,y) = (a \cn(u),\sn(u)) = (r \cos\theta,r \sin\theta) . $$ This angle $\theta$ is clearly not the same as the classically used angle $\phi = \am(u,k)$ which is such that $u = F(\phi,k)$ and $$ (\cn(u),\sn(u)) = (\cos\phi,\sin\phi) . $$ In fact, from these formulas it follows that $$ \tan\theta = \frac{\sn(u)}{a \cn(u)} = \frac{1}{a} \tan\phi . \tag{$*$} $$ So you're misquoting his definition by putting the angle $\phi$ where he actually uses $\theta$.
Schwalm defines $u$ as a function of $\theta$ through the differential relation $du = r \, d\theta$. Let us check that this does agree with the classical definition, where we have $$ \frac{du}{d\phi} = \frac{d}{d\phi} F(\phi,k) = \frac{1}{\sqrt{1 - k^2 \sin^2 \phi}} $$ from the fundamental theorem of calculus. To begin with, using $a^2 = 1/(1-k^2)$ we have $$ \begin{split} r^2 & = x^2 + y^2 = a^2 \cn^2(u) + \sn^2(u) \\ & = a^2 \cos^2 \phi + \sin^2 \phi = \frac{\cos^2 \phi}{1-k^2} + \sin^2 \phi = \frac{1 - k^2 \sin^2 \phi}{1-k^2} . \end{split} \tag{$**$} $$ Next, from the identity $(*)$ above, we obtain $$ \begin{aligned} \frac{d}{du} \tan\theta &= \frac{1}{a} \frac{d}{du} \tan\phi \\ \iff \quad (1 + \tan^2 \theta) \frac{d\theta}{du} &= \frac{1}{a} (1 + \tan^2 \phi) \frac{d\phi}{du} \\ \iff \quad (1 + a^{-2} \tan^2 \phi) \frac{d\theta}{du} &= \frac{1}{a \cos^2 \phi} \frac{d\phi}{du} \\ \iff \quad \frac{du}{d\theta} = \frac{1}{d\theta/du} & = (1 + a^{-2} \tan^2 \phi) (a \cos^2 \phi) \frac{1}{d\phi/du} \\ & = a (\cos^2 \phi + a^{-2} \sin^2 \phi) \frac{du}{d\phi} \\ & = (1-k^2)^{-1/2} (\cos^2 \phi + (1-k^2) \sin^2 \phi) \frac{1}{\sqrt{1 - k^2 \sin^2 \phi}} \\ & = \frac{\sqrt{1 - k^2 \sin^2 \phi}}{\sqrt{1-k^2}} = r , \end{aligned} $$ where $(**)$ was used in the very last step, so that indeed $du = r \, d\theta$, as claimed.
From $(*)$ we also get $a^2 \tan^2 \theta = \tan^2 \phi = \sin^2 \phi / (1 - \sin^2 \phi)$, which after a bit of calculation tells us that $1 - k^2 \sin^2 \phi = (1-k^2) / (1 - k^2 \cos^2 \theta)$, so that $$ r = \sqrt{\frac{1 - k^2 \sin^2 \phi}{1-k^2}} = \frac{1}{\sqrt{1 - k^2 \cos^2 \theta}} . $$ (This can also be seen, perhaps more easily, by inserting $(x,y)=(r \cos\theta, r\sin\theta)$ into the ellipse's equation $(x/a)^2+y^2=1$.) Thus, the relationship between $u$ and $\theta$ can be written as $$ u = \int_0^\theta \frac{d\alpha}{\sqrt{1 - k^2 \cos^2 \alpha}} , $$ to be contrasted with the relationship between $u$ and $\phi$, $$ u = F(\phi,k) = \int_0^\phi \frac{d\beta}{\sqrt{1 - k^2 \sin^2 \beta}} . $$
In the current version (Feb 2023) of the Wikipedia article for Jacobi elliptic functions, there is a section Definition as trigonometry: the Jacobi ellipse, which contains a construction involving the ellipse $x^2 + (y/b)^2 = 1$ with $b>1$. Despite the superficial similarity, and several references to Schwalm on the the article's talk page, this construction is not the same as the one in the video, since they write an integral producing a point $(x,y) = (r \cos\phi, r \sin\phi)$ on the ellipse, where $\phi$ (written $\varphi$ on Wikipedia) is the classical angle (not $\theta$), and then they project that point radially from the ellipse to the unit circle in order to get the point $(\cn u, \sn u) = (\cos\phi,\sin\phi)$. That is, they are not defining $\cn$ or $\sn$ directly as the $x$ or $y$ coordinate of some point on their ellipse, as Schwalm does with $\sn$ on his ellipse (he is projecting his point $(x,y)$ horizontally from the ellipse to the unit circle in order to get the point $(\cn u, \sn u) = (\cos\phi,\sin\phi)$).