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

By Eberhard Freitag, Rolf Busam

The assumption of this booklet is to offer an in depth description of the classical complicated research, right here ''classical'' potential approximately that sheaf theoretical and cohomological equipment are omitted.

The first 4 chapters conceal the basic center of advanced research featuring their primary effects. After this normal fabric, the authors breakthrough to elliptic features and to elliptic modular capabilities together with a flavor of all most pretty result of this box. The publication is rounded by way of functions to analytic quantity idea together with exclusive pearls of this interesting topic as for example the leading quantity Theorem. nice value is hooked up to completeness, all wanted notions are built, in simple terms minimum must haves (elementary evidence of calculus and algebra) are required.

More than four hundred workouts together with tricks for strategies and lots of figures make this an enticing, essential ebook for college students 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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Extra resources for Complex Analysis (2nd Edition) (Universitext)

Example text

If one decomposes f into real and imaginary parts, a f = u + iv , a+h then it follows that 1 f (a) = ∂1 u(a) + i∂1 v(a) = ∂2 u(a) + i∂2 v(a) . i From this the Cauchy-Riemann equations follow immediately. e. that the differentiability of f follows from the Cauchy-Riemann equations (with total differentiability assumed). It is well known that the existence of partial derivatives does not imply that f has a total derivative. But the following sufficient criterion for total differentiability is known from real analysis: If the partial derivatives of the map f : D −→ Rq , D ⊂ Rp open , exist at each point and are continuous, then f is totally differentiable.

Consider now the special case Az = l z , l = α + iβ , and thus A(x, y) = ( αx − βy , βx + αy ) , with z = (x, y) . This shows (1) ⇒ (4). The converse also follows from this formula. 5). Multiplication by r effects a dilation by the factor r, and multiplication by eiϕ gives a rotation by the angle ϕ. A map which is given by multiplication with a complex number is called also a similarity transformation. The maps of the complex plane C into itself, which can be written as multiplication with a nonzero complex number, are exactly the similarity transformations (rotation-dilations).

Remark. It is unsatisfying that the rather involved theorem of inverse functions of real analysis had to be used in the proof. A simple function theoretic proof therefore would be welcome. 6). Example. The exponential function exp is complex differentiable and its derivative does not vanish anywhere. The restriction of exp to the domain −π < Im z ≤ π is injective. However this region is not open. Thus we restrict exp to a somewhat smaller open region D := { z ∈ C ; −π < Im z < π } . Obviously we have exp(D) = C− = C \ { x ∈ R ; x ≤ 0 } (the complex plane slit along the negative real axis).

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