By E. Askwith

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**Extra resources for A Course of Pure Geometry**

**Example text**

In this way we also show the existence and uniqueness of the expansion of a proper fraction into partial fractions in the complex sense. Thus M(ClP'l) ':::::' C(z), where C(z) is the field of rational functions of one variable z. Hence meromorphic functions constitute a natural generalization of rational functions. Furthermore, these notions coincide for any compact Riemann surface, once the rationality of a function is suitably defined (see Sect. 5). Example 2. Let J: 3 1 -+ 3 2 be a nonconstant mapping of Riemann surfaces.

Then g(8) = ~n -1. Hence on any orient able compact surface there is a Riemann surface structure, a hyperelliptic one, for instance. Example 2. Let 8 be a compact Riemann surface of genus g, and 8 ---+

W2(t2). This form is called the tensor product of WI and W2' The tensor product of any number of differentials is defined in a similar way. Locally, every differential W can be written as a linear combination of tensor products of the differentials dz and dE, with functions as coefficients. If each product has i factors then W is said to be an i-form. Further, if all the coefficients are differentiable functions then we say that W is differentiable. As with multilinear functions, one can impose conditions of symmetry, skew- or Hermitian-symmetry, and the like.

### A Course of Pure Geometry by E. Askwith

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