By Bruce P. Palka

This e-book offers a rigorous but basic advent to the speculation of analytic features of a unmarried complicated variable. whereas presupposing in its readership a level of mathematical adulthood, it insists on no formal must haves past a valid wisdom of calculus. ranging from easy definitions, the textual content slowly and punctiliously develops the information of advanced research to the purpose the place such landmarks of the topic as Cauchy's theorem, the Riemann mapping theorem, and the concept of Mittag-Leffler could be handled with out sidestepping any problems with rigor. The emphasis all through is a geometrical one, so much mentioned within the huge bankruptcy facing conformal mapping, which quantities basically to a "short direction" in that very important zone of complicated functionality concept. every one bankruptcy concludes with a big variety of workouts, starting from elementary computations to difficulties of a extra conceptual and thought-provoking nature

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A,, We call B a diagonal matrix and write B = diag(hl. . , A,]). Then B has the characteristic equation Hence B has the eigenvalues h l , . . , A,,. We leave to Problem 5 below the discussion of the eigenvectors of B. We remark here that when h , . . h,, are n different numbers, the eigenvectors associated with the eigenvalue hk are the vectors ((0, . . 0. . , O), with 1 as kth entry and c nonzero. Sirrlilur matrices. Let A and B be n x n matrices. We say that B is similur to A if for some nonsingular n x n matrix C .

Au,. ,SO that Au4 = AUI and Aus = Au3. The simultaneous equations 3x + 2 y $ 5 2 = u , 4x - 5y - 82 = v , 7x + 2 y +9z = w are equivalent to the matrix equation '' for the product on the left-hand side equals the column vector [it P i;] and this equals col ( u , v , w) precisely when the given simultaneous equations hold. 56) can be replaced by the equations Au =y and B x = u. The elimination process at the beginning of this section is equivalent to replacing u by B x in the first equation to obtain y = A(Bx).

72). " By this we mean that in each row the first nonzero entry (the "leading entry") has column index which increases as the row index increases. An example is In the singular case. the first r rows (0 5 r 5 n - 1 ) contain nonzero entries, whereas the last n - r rows are all 0; here r is the rank of the matrix. In the nonsingular case r = n. By steps I, 11, and 111. 76) in which K has row echelon form and each leading entry is 1 and is the only nonzero entry in its column. The first r of the corresponding linear equations can now be solved for r of the unknowns in terms of thc rcrnaining n - r unknowns, which can be chosen arbitrarily.