By Serge Lang

ISBN-10: 0387962018

ISBN-13: 9780387962016

This 5th version of Lang's publication covers all of the issues ordinarily taught within the first-year calculus series. Divided into 5 elements, every one part of a primary direction IN CALCULUS comprises examples and purposes when it comes to the subject coated. furthermore, the rear of the publication includes unique options to a good number of the workouts, letting them be used as worked-out examples -- one of many major advancements over past variations.

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The resulting two-sheeted range of values for z is called a Riemann surface for the function 2^. The reader not familiar wifh these concepts should pay particular attention to the following exercises. EXERCISES 11. Describe the branch-point and branch-line situation for w = ZH. How many sheets does the Riemann surface have? Describe its topological character. Repeat for w = hi z and for w = [z(z — I)]**. (In this last case, write the formula for w as (r\r^ exp [i(2/2)], where r\ and r 2 are the distances of z from the two branch points and

Then for any point z, the corresponding value of w is uniquely defined—one branch value if z is on the lower plane, and the other if z is on the cone. The cone is of course now thought of as being flattened out so as to coincide with the first plane; the use of these two 24 Functions of a Complex Variable: Theory and Technique sheets for the z plane is essentially a topological device for making w singlevalued. The resulting two-sheeted range of values for z is called a Riemann surface for the function 2^.

Show, however, that if one enters another sheet of the Riemann surface by crossing this line, there will be a branch point at 2 = %• 13. Discuss the branch-cut and Riemann-surface situation for each of the following functions: where a is a constant with Re a = k > 0. In particular, verify that m(z) can define a function which has no branch points in Im z < k, that it can also define a function which has no branch points in Im 2 > — k, and that one linear combination of the two functions so defined is l/\/z* + a2.

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A first course in calculus by Serge Lang


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