newRPL: The complexity of complex mode

[Nigel (UK)]

A directed infinity might appear as:

• A limit - for example, the limit of a function as x tends to i times infinity;
• A limit of integration, if integration is to be carried out along a particular contour in the complex plane;
• The result of a limiting process - for example, the limit of 1/ix as x tends to zero from above.
  

I can't think of any other examples. In particular, it doesn't seem to me that the calculator needs to be able to manipulate directed infinities in any detail. It simply needs to have a representation of them that allows them to be used as inputs to various CAS routines and to be displayed as an output.

I suggest two types of infinity - complex infinity, and directed infinity \(\infty\). Directed infinity satisfies the following rules:

• \(+\infty\) and \(-\infty\) are particular cases of directed infinity.
• Directed infinity is represented as \(e^{i \theta}\times\infty\) (polar form) or as \((\cos\theta+i\sin\theta)\times\infty\) (rectangular form). (Using \(\infty\) with an angle following is fine as well, if that looks better on the calculator.)
• \(z\times\infty\) is just \(e^{i\times\arg(z)}\times\infty\): only the direction matters, not the magnitude.
• \(e^{i \theta}\times\infty+z=e^{i \theta}\times\infty\) for all complex \(z\).
• \(z_1\infty+z_2\infty\) is only meaningful if \(\arg(z_1)=\arg(z_2)\), otherwise it is indeterminate. I'd prefer indeterminate to complex infinity; I really feel that if someone is trying to add two different directed infinities the calculator should do nothing to encourage them.
• Multiplication by \(\infty\) does not distribute over addition, so \((\cos\theta+i\sin\theta)\times\infty\) does not expand into the indeterminate \(\cos\theta\times\infty+i\sin\theta\times\infty\).
  

These rules, I think, agree with everything in your first post. Going through your list of examples:

• \((+\infty,0)=(+\infty,3)=+\infty\) as adding either zero or \(3i\) doesn't change the value of \(+\infty\).
• \((-\infty,0)=-\infty+0i=-\infty\) by the same reasoning.
• \((0,+\infty)\) or \((0,-\infty)\) are fine: they are \(\pm i\infty\) or \(e^{\pm i\pi/2}\infty\) or \(\infty\angle\pm90^\circ\) as preferred.
• \((\infty,\infty)\) is indeterminate, because it is the addition of two different directed infinities. If someone requires an infinity directed at \(45^\circ\) they should enter \((\cos 45^\circ+i\sin 45^\circ)\infty\) or \(e^{i\pi/4}\infty\) or \(\infty\angle 45^\circ\).
  
  

Does this help?

Nigel (UK)