Complex Analysis (2nd Edition) (Universitext) by Eberhard Freitag, Rolf Busam

By Eberhard Freitag, Rolf Busam

The belief of this publication is to provide an intensive description of the classical complicated research, right here ''classical'' ability approximately that sheaf theoretical and cohomological equipment are omitted.

The first 4 chapters conceal the basic center of complicated research featuring their basic effects. After this common fabric, the authors breakthrough to elliptic features and to elliptic modular features together with a flavor of all most lovely result of this box. The publication is rounded through purposes to analytic quantity thought together with exceptional pearls of this attention-grabbing topic as for example the top quantity Theorem. nice value is connected to completeness, all wanted notions are constructed, merely minimum necessities (elementary evidence of calculus and algebra) are required.

More than four hundred workouts together with tricks for ideas and lots of figures make this an enticing, vital e-book for college kids who wish to have a legitimate advent to classical advanced analysis.

For the second one version the authors have revised the textual content conscientiously.

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Show: A sequence (zn )n≥0 , zn ∈ C is convergent if and only if it is a Cauchy sequence. I Differential Calculus in the Complex Plane C 32 4. Prove the following inequalities. (a) For all z ∈ C we have |exp(z) − 1| ≤ exp(|z|) − 1 ≤ |z| exp(|z|) . (b) For all z ∈ C with |z| ≤ 1 we have |exp(z) − 1| ≤ 2 |z| . 5. Determine, in each case, all the z ∈ C with exp(z) = −2 , 6. exp(z) = −i , exp(z) = i , sin z = 100 , sin z = 7i , sin z = 1 − i , cos z = 3i , cos z = 3 + 4i , cos z = 13 . The (complex) hyperbolic functions cosh and sinh are defined similarly to the real ones.

1. 2. 2 show that Arg : C− → R is continuous. Here C− is the complex plane slit along the negative real axis : C− := C \ { t ∈ R ; t ≤ 0 } . Deduce that the principal value of the logarithm is also continuous on C− . 40 I Differential Calculus in the Complex Plane C 3. Set D ⊂ Rp . A point a ∈ D is called an interior point (of D) if together with a there exists a ε-ball Uε (a) := { x ∈ Rp ; |x − a| < ε } which is contained in D. Show: D is open ⇐⇒ each point of D is an interior point. A subset U ⊂ Rp is called a neighborhood of a ∈ Rp if U contains an ε-ball Uε (a).

E. sinh(z + 2πi) = sinh z cosh(z + 2πi) = cosh z (e) For all z ∈ C the series z 2n and (2n)! for all z ∈ C . z 2n+1 are absolutely conver(2n + 1)! gent, and one has ∞ cosh z = n=0 7. z 2n (2n)! ∞ and sinh z = n=0 z 2n+1 . (2n + 1)! For all z = x + iy ∈ C one has: (a) (b) exp(z) = exp(z) , sin(z) = sin(z) , cos(z) = cos(z) . cos z = cos(x + iy) = cos x cosh y − i sin x sinh y , sin z = sin(x + iy) = sin x cosh y + i cos x sinh y . 2 Convergent Sequences and Series 33 In the special case x = 0, y ∈ R we have cos(iy) = 1 y (e + e−y ) = cosh y 2 and sin(iy) = i y (e − e−y ) = i sinh y .

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