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The complex plane C with the origin removed, i.e., C-{0}. The punctured plane is sometimes denoted C^* (although this notation conflicts with that for the Riemann sphere C-*, ...

The Riemann sphere, also called the extended complex plane, is a one-dimensional complex manifold C^* (C-star) which is the one-point compactification of the complex numbers ...

There are a couple of versions of this theorem. Basically, it says that any bounded linear functional T on the space of compactly supported continuous functions on X is the ...

The complex conjugate of a complex number z=a+bi is defined to be z^_=a-bi. (1) The conjugate matrix of a matrix A=(a_(ij)) is the matrix obtained by replacing each element ...

The modulus of a complex number z, also called the complex norm, is denoted |z| and defined by |x+iy|=sqrt(x^2+y^2). (1) If z is expressed as a complex exponential (i.e., a ...

The representation, beloved of engineers and physicists, of a complex number in terms of a complex exponential x+iy=|z|e^(iphi), (1) where i (called j by engineers) is the ...

A complex function is said to be analytic on a region R if it is complex differentiable at every point in R. The terms holomorphic function, differentiable function, and ...

A gadget defined for complex vector bundles. The Chern classes of a complex manifold are the Chern classes of its tangent bundle. The ith Chern class is an obstruction to the ...

In a cochain complex of modules ...->C^(i-1)->^(d^(i-1))C^i->^(d^i)C^(i+1)->..., the module B^i of i-coboundaries is the image of d^(i-1). It is a submodule of C^i and is ...

A complex manifold is a manifold M whose coordinate charts are open subsets of C^n and the transition functions between charts are holomorphic functions. Naturally, a complex ...