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If f(z) is regular and of the form O(e^(k|z|)) where k<pi, for R[z]>=0, and if f(z)=0 for z=0, 1, ..., then f(z) is identically zero.

The winding number of a contour gamma about a point z_0, denoted n(gamma,z_0), is defined by n(gamma,z_0)=1/(2pii)∮_gamma(dz)/(z-z_0) and gives the number of times gamma ...

A hyperfunction, discovered by Mikio Sato in 1958, is defined as a pair of holomorphic functions (f,g) which are separated by a boundary gamma. If gamma is taken to be a ...

Each double point assigned to an irreducible algebraic curve whose curve genus is nonnegative imposes exactly one condition.

If a function analytic at the origin has no singularities other than poles for finite x, and if we can choose a sequence of contours C_m about z=0 tending to infinity such ...

A multivalued function, also known as a multiple-valued function (Knopp 1996, part 1 p. 103), is a "function" that assumes two or more distinct values in its range for at ...

The unique nonnegative square root of a nonnegative real number. For example, the principal square root of 9 is 3, although both -3 and 3 are square roots of 9. The concept ...

A conformal mapping from the upper half-plane to a polygon.

The sign of a real number, also called sgn or signum, is -1 for a negative number (i.e., one with a minus sign "-"), 0 for the number zero, or +1 for a positive number (i.e., ...

The three circles theorem, also called Hadamard's three circles theorem (Edwards 2001, p. 187), states that if f is an analytic function in the annulus 0<r_1<|z|<r_2<infty, ...