Complex infinity

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complexInfinity represents the only non-complex point of the one-point compactification of the complex numbers.

Mathematically, complexInfinity is the north pole of the Riemann sphere, with the unit circle as equator and the point 0 at the south pole.

With respect to arithmetic, complexInfinity behaves like “1/0”. In particular, non-zero complex numbers may be multiplied or divided by complexInfinity or 1/ complexInfinity. Adding complexInfinity to a finite number yields again complexInfinity.

With respect to arithmetical operations, complexInfinity is incompatible with the real infinity.


Example 1

complexInfinity can be used in arithmetical operations with complex numbers. The result in multiplications or divisions is either complexInfinity, 0, or undefined:

3*complexInfinity, I*complexInfinity, 0*complexInfinity;
3/complexInfinity, I/complexInfinity, 0/complexInfinity;
complexInfinity/3, complexInfinity/I;
complexInfinity*complexInfinity, complexInfinity/complexInfinity;

The result in additions is undefined if one of the operands is infinite, and complexInfinity otherwise:

complexInfinity + complexInfinity, infinity + complexInfinity;
3 + complexInfinity, I + complexInfinity, PI + complexInfinity

Symbolic expressions in arithmetical operations involving complexInfinity are implicitly assumed to be different from both 0 and complexInfinity:

delete x:
x*complexInfinity, x/complexInfinity, complexInfinity/x,
x + complexInfinity


complexInfinity is the only element of the domain stdlib::CInfinity.

See Also

MuPAD Functions