# Download e-book for kindle: A Course in Complex Analysis: From Basic Results to Advanced by Wolfgang Fischer, Ingo Lieb, Jan Cannizzo

By Wolfgang Fischer, Ingo Lieb, Jan Cannizzo

ISBN-10: 3834815764

ISBN-13: 9783834815767

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Additional info for A Course in Complex Analysis: From Basic Results to Advanced Topics

Example text

We write: Δν (z0 ) = ∂f (z0 ) = fzν (z0 ). ∂zν If f is holomorphic on all of U , the maps z→ ∂f (z), ∂zν z ∈ U, 6. Several complex variables 33 deﬁne the complex partial derivatives of f as functions on U ; we denote them by fzν = ∂f . +kn f , . . , k1 . 2, namely in terms of the relation n n Δν (z)(zν − zν0 ) + f (z) − f (z0 ) = ν=1 Eν (z)(z ν − z 0ν ), (2) ν=1 where Δν and Eν are continuous at z0 , then as before we conclude: The values Δν (z0 ) and Eν (z0 ) are uniquely determined by f and z0 .

Show that f (z) = f (z) for all z ∈ G. b) Suppose G = Dr (0) and f is holomorphic on G and real-valued on G ∩ R. Show that if f is even (odd), then the values of f on G ∩ iR are real (imaginary). Prove this without using the power series expansion of f . 7. a) Suppose the domain G is symmetric with respect to the real axis and f is continuous on G and holomorphic on G \ R. Show that f is holomorphic on all of G. Hints: Use Morera’s theorem. In splitting up the triangle Δ ⊂ G, one sees that the only problematic case is the one in which an edge of Δ lies on R.

4. Elementary functions 23 Proof: Let us restrict our attention to the cosine function. The equation cos z = 1 iz (e + e−iz ) = 0 2 implies that e2iz + 1 = 0. (12) The only solutions of (12) are the aforementioned ones. In real analysis, one sometimes deﬁnes π via the condition that π/2 is the smallest positive zero of the cosine function. The above theorem shows that our deﬁnition of π gives the same number as this more elementary deﬁnition. The connection between π and the circumference of a circle will be derived in the following section.