Complex Variables by Robert B. Ash (Auth.) PDF By Robert B. Ash (Auth.)

ISBN-10: 0120652501

ISBN-13: 9780120652501

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Example text

Be an arbitrary sequence of complex numbers. (a) If lim sup a n+l = a, what conclusions can be drawn about the radius of convergence of the power series Σ #n zn ? n=0 37 POWER SERIES (b) approaches a limit a, what conclusions can be If drawn? 4. , where ( i n ) 1 / n - * ° o as n—> oo. 5. Let Rn{z) be the remainder after the term of degree n in the Taylor expansion of a function/about z 0 . 14. , show that I Λη(ζ)| < A (-£-) , where Λ = Μ,(Γ) —^— . (Summation by Parts) Let {an} and {èn} be sequences of complex numbers.

Furthermore, the integer is independent of the particular continuous argument chosen. PROOF For each / e [a, 6], ρίβ«) _ y(0 I r(0l Thus ρΐ(β(ί>)-9(α)) Y(b) \γφ)\ I γ(α)\ γ(α) 55 THE INDEX OF A POINT since y is closed, and it follows that -=— [0(b) — 6(a)] is an integer. 2(c). Thus 9(b) = φ(ο) + 2-nm, θ(ά) = φ(α) + Ιπνη, and the result follows. 3 Definition Let y be a closed curve, and z0 a point not in y*. Denote by y + w the curve y(t) + w, a < / < b. Let Θ be a continuous argument of y — z 0 .

6; since y* n (^4 u B) = 0 , we have y* C [/, that is, y is a cycle in U. 6, we may choose the mesh so that one of the squares is centered at a particular z e A; then η(γ, z) = 1, z G AC C — U, contradicting (2). 4. 9. /' (4) implies (5): If/ is analytic and never 0 on U, -j- is analytic on U, hence has a primitive on U. 5, / h a s an analytic logarithm.