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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.2012-09-10

1 Answers 1

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I was too fast in asking the question. Further reading demystified the nature of the separators. Sorry for my impatience. I quote from Wikipedia:

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 $F(\varphi, \sin \alpha) = F(\varphi \,|\, \sin^2 \alpha) = F(\varphi \setminus \alpha) = F(\sin \varphi ; \sin \alpha)$