# An Introduction to Complex Function Theory by Bruce P. Palka

By Bruce P. Palka

This e-book presents a rigorous but easy creation 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 necessities past a valid wisdom of calculus. ranging from easy definitions, the textual content slowly and thoroughly develops the guidelines of complicated 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 might be handled with out sidestepping any problems with rigor. The emphasis all through is a geometrical one, such a lot suggested within the wide bankruptcy facing conformal mapping, which quantities primarily to a "short direction" in that very important quarter of complicated functionality concept. every one bankruptcy concludes with a big variety of workouts, starting from undemanding computations to difficulties of a extra conceptual and thought-provoking nature.

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The two transmitters send out signals simultaneously to a ship that is located at P. Suppose that the ship receives the signal from B, 800 microseconds (m sec) before it receives the signal from A. a. Assuming that radio waves travel at a speed of 980 ft/m sec, find an equation of the hyperbola on which the ship lies (see page 842). Hint: d(P, A) Ϫ d(P, B) ϭ 2a. 1 Conic Sections b. If the ship is sailing in a direction parallel to and 20 mi north of the coastline, locate the position of the ship at that instant of time.

X ϭ sec u, y ϭ tan u; Ϫp2 Ͻ u Ͻ p2 24. x ϭ et, y ϭ sin u Ϫ 2; 0 Յ u Յ 2p 16. x ϭ sec u, y ϭ cos u 22. x ϭ t 3, yϭtϩ1 12. x ϭ cos u ϩ 1, 15. x ϭ cos u, y ϭ 3 cos u ϩ 1; 19. x ϭ sin2 u, y ϭ sin4 u; 0 Յ u Յ p2 0ՅtՅ3 6. x ϭ t 3, 14. x ϭ sin u ϩ 3, 0 Յ u Յ 2p y ϭ 3 sin u Ϫ 1; y ϭ ln t 25. x ϭ cosh t, y ϭ sinh t 26. x ϭ 3 sinh t, y ϭ 2 cosh t 2 27.

5 y ϭ Ϫ√3x 839 2 x EXAMPLE 8 A Rutherford Scattering A massive atomic nucleus used as a target for incoming alpha particles is located at the point (Ϫ2, 0), as shown in Figure 25. Suppose that an alpha particle approaching the nucleus has a trajectory that is a branch of the hyperbola shown with asymptotes y ϭ Ϯ 13x and foci (Ϯ2, 0). Find an equation of the trajectory. Solution The asymptotes of a hyperbola with center at the origin and foci lying on the x-axis have equations of the form y ϭ Ϯ(b>a)x.