What does | mean?

I found this symbol on Wolfram|Alpha. Does it mean "or"?

$$ \large \cos^{-1}(-1)=\mathrm{cd}^{-1}(-1\mid 0)$$

Solution 1:

Since I don't believe any of the previous two answers explained the notation properly (which I guess is forgivable since knowledge of elliptic stuff is not as common as it once was), here's my take:

The funny thing about the theory of elliptic integrals and elliptic functions is that people use related, but rather completely different conventions and notation. One guy has his favorite set, one has another way of doing things; heck, I have a particular preference myself. I'll explain three of them.

The incomplete elliptic integral of the first kind can be defined in at least three ways:

$$\begin{align*} F(\phi,k)&=\int_0^\phi \frac{\mathrm du}{\sqrt{1-k^2\sin^2 u}}\\ F(\phi\mid m)&=\int_0^\phi \frac{\mathrm du}{\sqrt{1-m\,\sin^2 u}}\\ F(\phi\backslash \alpha)&=\int_0^\phi \frac{\mathrm du}{\sqrt{1-\sin^2\alpha\,\sin^2 u}} \end{align*}$$

Yes, kids, this is a Spot The Difference game! If you compare these three definitions, we have the relationship

$$m=k^2=\sin^2\alpha$$

Now, $m$ is what's called a parameter; $k$ is what's called a modulus; and, $\alpha$ is termed the modular angle. Whichever of these arguments one is concerned with in elliptic integrals is easily indicated by the choice of delimiter: comma for modulus, pipe/bar for parameter, and backslash for modular angle.

Since Jacobian elliptic functions can be constructed in terms of the inverse of the incomplete elliptic integral of the first kind (what is called the Jacobi amplitude, $\mathrm{am}(u,k)$/$\mathrm{am}(u\mid m)$/$\mathrm{am}(u\backslash \alpha)$), and since the inverse Jacobian elliptic functions can be expressed as compositions of the incomplete elliptic integral of the first kind with inverse trigonometric functions (in particular, we have $\mathrm{cd}^{-1}(w\mid m)=F\left(\dfrac{\pi}{2}\mid m\right)-F(\arcsin\,w\mid m)$), they too inherit the delimiter convention used for elliptic integrals. (Yes, it's a rather confusing system, but there you are.)

So that's it: Wolfram Alpha likes using the parameter convention for its elliptic integrals and elliptic functions, which is why you see the nice pipe cleanly separating the arguments of the inverse Jacobian elliptic function in Alpha's output.

Solution 2:

It is called a vertical bar or pipe.

Divisibility: $a|b$ could mean $a$ divides $b$.
In programming, $||$ means boolean or. $|$ means bitwise or.
In the case of the Jacobi Elliptic Function, it separates the two arguments $v$ and $m$.

You could look it up on Wikipedia or MathWorld.

Solution 3:

Flawed Guess. Look at J.M.'s Answer

It is the inverse Jacobi Elliptic function.

You could well ask Prof. Google for more links. I would not want to list all of them here.

As people have commented below, the $\vert$ by itself means nothing. It just separates the two arguments of the function $\mathrm{cd}^{-1}$. And, $\mathrm{cd}^{-1}$ is a fancy way of writing, in this context, $\mathrm{cd}^{-1}(x|k)=\arccos(x,k)$.

As other answers have pointed out, this symbol means many things.