exists. For example, the function f(z) = exp(1/z) has an essential singularity at z0 = 0. |
exists. For example, the function f(z) = exp(1/z) (see exponential function) has an essential singularity at z0 = 0. |
The Weierstrass-Casorati theorem states that if f has an essential singularity at z0, and V is any neighborhood of z0 contained in U, then f(V) is dense in C. Or spelled out: if δ > 0 and w is any complex number, then there exists a complex number z in U with |z - z0| < δ and |f(z) - w| < δ. |
The Weierstrass-Casorati theorem states that if f has an essential singularity at z0, and V is any neighborhood of z0 contained in U, then f(V) is dense in C. Or spelled out: if ε > 0 and w is any complex number, then there exists a complex number z in U with |z - z0| < ε and |f(z) - w| < ε. |
Start with an open subset U of the complex plane containing the number z0, and a holomorphic function f defined on U - {z0}. The complex number z0 is called an essential singularity if there is no natural number n such that the limit
lim f(z)·(z-z0)n z→z0
exists. For example, the function f(z) = exp(1/z) (see exponential function) has an essential singularity at z0 = 0.
The Weierstrass-Casorati theorem states that if f has an essential singularity at z0, and V is any neighborhood of z0 contained in U, then f(V) is dense in C. Or spelled out: if ε > 0 and w is any complex number, then there exists a complex number z in U with |z - z0| < ε and |f(z) - w| < ε.