Download Complex Variables: Introduction and Applications by Ablowitz M.J., Fokas A.S. PDF

By Ablowitz M.J., Fokas A.S.

Advanced variables supply very effective tools for attacking many tricky difficulties, and it's the goal of this e-book to provide a radical evaluation of those equipment and their purposes. half I is an advent to the topic, together with residue calculus and remodel tools. half II advances to conformal mappings, and the research of Riemann-Hilbert difficulties. an in depth array of examples and workouts are integrated. This re-creation has been more desirable all through and is perfect to be used in introductory undergraduate and graduate point classes in advanced variables. First variation Hb (1997): 0-521-48058-2 First version Pb (1997): 0-521-48523-1

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N=1 n=0 θp = 0 x θp = 2π Fig. 3. Logarithm function and branch cut 50 2 Analytic Functions and Integration n=1 θp = π n=0 -x θ p = −π n=2... Fig. 4. Logarithm function and alternative branch cut N N iy iy image image x S x -x S (a) Re z > 0 (b) Re z < 0 Fig. 5. 4). It should be noted that the point z = ∞ is also a branch point for log z. As we have seen, the point at infinity is easily understood via the transformation z = 1/t, so that t near zero corresponds to z near ∞. The above arguments, which are used to establish whether a point is in fact a branch point, apply at t = 0.

That is, they are harmonic functions in D, and v is referred to as the harmonic conjugate of u (and vice versa). The function v may be obtained from u via the Cauchy–Riemann conditions. It is clear from the derivation of Eqs. 11a,b) that f (z) = u(x, y) + iv(x, y) is an analytic function if and only if u and v satisfy Eqs. 11a,b) and v is the harmonic conjugate of u. The following example illustrates how, given u(x, y), it is possible to obtain the harmonic conjugate v(x, y) as well as the analytic function f (z).

The associated boundary conditions are as follows. , the normal velocity) must vanish on a rigid boundary of an ideal fluid. Because we have shown that the level sets φ(x, y) = constant and ψ(x, y) = const. are mutually orthogonal at any point (x, y), we conclude that the level sets of the stream function ψ follow the direction of the flow field; namely, they follow the direction of the gradient of φ, which are themselves orthogonal to the level sets of φ. The level curves ψ(x, y) = const. are called streamlines of the flow.

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