# What does | mean?

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

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

• You must have been able to see the documentation for it there! – user21436 Feb 12 '12 at 19:00
• In this case it is just separating the $2$ arguments of the inverse Jacobi elliptic function, as you can find by clicking links from Wolfram alpha when this equation appears. E.g. reference.wolfram.com/mathematica/ref/InverseJacobiCD.html – Jonas Meyer Feb 12 '12 at 19:06
• NAND operator – Peđa Terzić Feb 12 '12 at 19:11
• I know that it isn't the same meaning you're using, but I have seen the vertical bar be used to indicate "such that" when defining a set. For instance, $A=\{x\in N | x>3\}$ would be the set A of naturals such that each element is greater than 3. I'm not sure how common that is, though. – preferred_anon Jul 29 '12 at 13:20

## 3 Answers

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.

• More general, $||$ means the OR operation in any high level programming language I can think of. – Hidde Feb 13 '12 at 6:29
• Also used in conditional probabilities. – abnry Mar 3 '15 at 0:38
• @Hidde Does it mean OR in Pascal? SQL? XQuery? – CiaPan Nov 2 '15 at 15:32

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.

• Interesting, +1! – user21436 Feb 13 '12 at 6:08

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.

• $a|b \iff$ $a$ divides $b$.

For other meanings in Logic, look up Wikipedia.

• I meant the symbol $|$ . I just wanted to give you some context. – David Feb 12 '12 at 19:06
• @David: The symbol $|$ is only separating the $2$ arguments of the function in this case. – Jonas Meyer Feb 12 '12 at 19:08
• @Jonas Beat me to it! I don't see any other meaning to it! – user21436 Feb 12 '12 at 19:10
• The symbol | has dozens of meanings in mathematics. In this case, it looks like it doesn't mean anything by itself; $\operatorname{cd}^{-1} (x | k)$ is just another way of writing $\operatorname{arccd} (x, k)$. – Tanner Swett Feb 12 '12 at 19:11
• @Tanner and Kanappan: surprisingly enough, the choice of delimiter displayed does matter. See my answer. – J. M. is a poor mathematician Feb 13 '12 at 6:02