The bestknown properties and formulas for symbols
Specific values
The symbol has the following values at some finite points :
The symbol has the following values at some infinite points :
The symbol has the following value at point :
General characteristics
is a symbol. It represents an unknown or not exactly determined point (potentially with infinite magnitude) of the complex plane. Often it results from a double limit where two infinitesimal parameters approach zero at different speeds (e.g. ).
is a symbol. On the Riemann sphere, it is the north pole approached from exactly east. In the projective complex plane, it is a point at the line at infinity.
is a symbol. On the Riemann sphere, it is the north pole. In the projective complex plane, it is the line at infinity.
is a symbol. On the Riemann sphere, it is the north pole together with the direction how to approach it. In the projective complex plane, it is a point at the line at infinity.
Complex characteristics
The symbols , , , and have the following complex characteristics:
Differentiation
Derivatives of the symbols , , , and satisfy the following relations:
Integration
Simple indefinite integrals of the symbols , , and have the following representations:
Integral transforms
All Fourier integral transforms of the symbols , , , and can be evaluated using the following formal rules:
Laplace direct and inverse integral transforms of the symbols , , , and can be evaluated using the following formal rules:
Inequalities
The symbols , , and satisfy some obvious inequalities, for example:
