Higher Dimensions
In several complex variables, a meromorphic function is defined to be locally a quotient of two holomorphic functions. For example, is a meromorphic function on the two-dimensional complex affine space. Here it is no longer true that every meromorphic function can be regarded as holomorphic function with values in the Riemann sphere: There is a set of "indeterminacy" of codimension two (in the given example this set consists of the origin ).
Unlike in dimension one, in higher dimensions there do exist complex manifolds on which there are no non-constant meromorphic functions, for example, most complex tori.
Read more about this topic: Meromorphic Function
Famous quotes containing the words higher and/or dimensions:
“We are conscious of an animal in us, which awakens in proportion as our higher nature slumbers. It is reptile and sensual, and perhaps cannot be wholly expelled; like the worms which, even in life and health, occupy our bodies. Possibly we may withdraw from it, but never change its nature. I fear that it may enjoy a certain health of its own; that we may be well, yet not pure.”
—Henry David Thoreau (1817–1862)
“The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”
—Thomas Jefferson (1743–1826)