By Theodore Gamelin

Gamelin's e-book covers a fascinating and wide selection of issues in a just a little unorthodox demeanour. Examples: Riemann surfaces are brought within the first bankruptcy, while winding numbers don't make an visual appeal until eventually midway into the booklet. Cauchy's theorem and its family are as an alternative built within the context of piecewise-smooth obstacles of domain names (in specific, basic closed curves) and purely later generalized to arbitrary closed paths, nearly as an afterthought.

In common, the writer effectively conveys the spirit of the topic, and manages to take action particularly successfully. It's now not the main painstakingly rigorous textual content available in the market, and the reader is anticipated to fill in the various information himself, however the payoff is lot of flooring is roofed with no getting slowed down in technicalities. in lots of books in this topic it may be difficult to determine the wooded area for the bushes. This one is a delightful exception.

There are loads of reliable advanced research books available in the market: Conway, Ahlfors, Remmert, Palka, Narasimhan, the second one half enormous Rudin, and naturally Needham's "Visual complicated Analysis." (And many others which are well-regarded yet that i haven't checked out, similar to Lang and Jones/Singerman, in addition to the previous classics via Hille, Knopp, Cartan, Saks and Zygmund.) almost all these has its personal standpoint, and complicated research is a huge, multifaceted topic that's maybe top studied from a number of issues of view. an individual eager to research this topic good will reap the benefits of having numerous books at hand.

Gamelin's contribution to the pantheon isn't really progressive, however it does gather among its pages a large collection of themes now not mostly present in a unmarried textual content. The reader is whisked from the fundamentals to the Riemann mapping theorem in three hundred pages with dazzling ease.

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**Additional info for Complex Analysis (Undergraduate Texts in Mathematics)**

**Example text**

This is equivalent to z = x+iy. 2) arge Z = y. If z is real (y = 0), the definition of eZ agrees with the usual exponential function eX. If z is imaginary (x = 0), the definition agrees with the definition of ei9 given in Section 2. 20 I The Complex Plane and Elementary Functions A fundamental property of the exponential function is that it is periodic. The complex number,x is a period of the function J(z) if J(z +,x) = J(z) for all z for which J(z) and J(z + ,x) are defined. The function J(z) is periodic if it has a nonzero period.

This identity is to be understood as a set identity, in the sense that w satisfies sin w = z if and only if w is one of the values of -i log (iz ± v'f=Z2). To obtain a genuine function, we must restrict the domain and specify the branch. One way to do this is to draw two branch cuts, from -00 to -1 and from + 1 to +00 along the real axis, and to specify the branch of JI=Z2 that is positive on the interval (-1,1). With this branch of J1- z2, we obtain a continuous branch -iLog (iz + v'f=Z2) of sin- 1 z.

Each w E C\{O} corresponds to exactly two points on the surface. The function f(w) on the surface represents the multivalued function y'w in the sense that the values of y'w are precisely the values assumed by f(w) at the points of the surface lying over w. The surface we have constructed is called the Riemann surface of y'w. The surface is essentially a sphere with two punctures corresponding to 0 and 00. One way to see this is to note that the function f(w) maps the surface one-to-one onto the z-plane punctured at O.