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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 ...

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 ...

A Hermitian inner product on a complex vector space V is a complex-valued bilinear form on V which is antilinear in the second slot, and is positive definite. That is, it ...

In a chain complex of modules ...->C_(i+1)->^(d_(i+1))C_i->^(d_i)C_(i-1)->..., the module B_i of i-boundaries is the image of d_(i+1). It is a submodule of C_i and is ...

A complex manifold for which the exterior derivative of the fundamental form Omega associated with the given Hermitian metric vanishes, so dOmega=0. In other words, it is a ...

The real part R[z] of a complex number z=x+iy is the real number not multiplying i, so R[x+iy]=x. In terms of z itself, R[z]=1/2(z+z^_), where z^_ is the complex conjugate of ...