# What do those different separators of arguments mean?

While studying the wikipedia article about elliptic integrals, I encountered this notation, which I do not understand.

$$F(\varphi,k) = F(\varphi \,|\, k^2) = F(\sin \varphi ; k) = \int_0^\varphi \frac {d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}$$

What do those separators

• $F(\cdot,\cdot)$
• $F(\cdot|\cdot)$
• $F(\cdot;\cdot)$

of the arguments of $F$ mean explicitly?

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It's a way of avoiding confusion when you have a family of functions that can be parametrised in different ways. –  Zhen Lin Sep 10 '12 at 13:17
In this notation, the use of a vertical bar as delimiter indicates that the argument following it is the "parameter", while the backslash indicates that it is the modular angle. The use of a semicolon implies that the argument preceding it is the sine of the amplitude. So $$F(\varphi, \sin \alpha) = F(\varphi \,|\, \sin^2 \alpha) = F(\varphi \setminus \alpha) = F(\sin \varphi ; \sin \alpha)$$