Complex Analysis (Undergraduate Texts in Mathematics)
Language: English
Pages: 478
ISBN: 0387950699
Format: PDF / Kindle (mobi) / ePub
An introduction to complex analysis for students with some knowledge of complex numbers from high school. It contains sixteen chapters, the first eleven of which are aimed at an upper division undergraduate audience. The remaining five chapters are designed to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis. Topics studied include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces, with emphasis placed on the three geometries: spherical, euclidean, and hyperbolic. Throughout, exercises range from the very simple to the challenging. The book is based on lectures given by the author at several universities, including UCLA, Brown University, La Plata, Buenos Aires, and the Universidad Autonomo de Valencia, Spain.
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fourth condition above) means simply that this integral is zero for all small c > O. This occurs if and only if the integrand is zero, that is, ap 8y (6.3) aQ ax· Thus the irrotationality of the flow on D means that (6.3) holds on D. The mathematical formulation of the second and third conditions is that the net flow of fluid across the boundary of any small circle 'Ye centered at a point Zo of D is zero. Using Green's theorem again, we obtain 0= 1 h. Y·nds = 1 Pdy - Qdx = "IE
(conformal map) from D onto a domain U, and if fo(w) = 4>o(w) + i1j;o(w) is a complex velocity potential for a flow on U, then the composition f(z) = fo(h(z» is analytic, hence the complex velocity potential for a flow on D. One of the simplest flows to understand is the constant horizontal flow on the upper half-plane ]HI = {1m z > O}, for which no fluid enters or leaves across the bounding real line JR. A complex velocity potential for the flow is 6. Applications to Fluid Dynamics 95 fo(z)
the length of'Y is The notation can be justified by considering the sums approximating these integrals. For the subdivision of the parameter interval used earlier, the usual sum used in multivariable calculus to approximate h(x, y) ds is i i7 h(x, y) ds ~ L h(xj, Yj)V(Xj+1 - Xj)2 + (Yj+1 - Yj)2. In complex notation this becomes (1.4) In particular, the sum approximating the length L of 'Y is given by (1.5) L ~ L IZj+1 - zjl· Example. The parameterization z(B) = zo+Re i8 of the circle
inequality i zn 21fRn+l ---dz < - - Izl=R zm - 1 Rm - 1 ' R > 1, m ~ 1, n ~ O. 2. Fundamental Theorem of Calculus for Analytic Functions 107 8. Suppose the continuous function J (e ill ) on the unit circle satisfies IJ(eill)1 ~ M and I ~zl=l J(z)dzl = 2rrM. Show that J(z) = cZ for some constant c with modulus lei = M. 9. Suppose h(z) is a continuous function on a curve ,. Show that H(w) = 1 '"f h(z) dz, z-w is analytic on the complement of " and find H' (w). 2. Fundamental Theorem
by multiplying the polynomials and gathering terms, where here the notation O(zm) is used for terms involving powers zk for k ~ m. (This is consistent with our earlier use of the "big-oh" notation. See Exercise VI.2.6.) For k ::; n the coefficient of zk in this polynomial is exactly Ck given by (6.1). Passing to the limit as n --+ 00, we find that f(z)g(z) also has power series coefficient of zk equal to Ck. The power series of a quotient f (z) / g( z) can also be effectively computed. It
