This is a collection of exercises in the theory of analytic functions, with completed and detailed solutions. We wish to introduce the student to applications and aspects of the theory of analytic functions not always touched upon in a first course. Using appropriate exercises we wish to show to the students some aspects of what lies beyond a first course in complex variables. We also discuss topics of interest for electrical engineering students (for instance, the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). Examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space) are given. The book also includes a part where relevant facts from topology, functional analysis and Lebesgue integration are reviewed.

third equality and also using the third equality itself we have n−1 (z − exp p(z) = k=0 −ikπ ikπ )(z − exp )= n n n−1 (z 2 − 2zRe zk + 1). k=1 We now prove (1.5.9). Setting z = 1 in the above equality we have n−1 (2 − 2 cos( n= k=1 Recall that 1 − cos( kπ )). n kπ kπ ) = 2 sin2 ( ). n 2n Hence n−1 (2 − 2 cos( n= k=1 n−1 4 sin2 ( = k=1 kπ ) 2n n−1 sin2 ( = 4n−1 kπ )) n k=1 kπ ) 2n and hence the result by taking the square root of both sides since the numbers kπ sin( ) > 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 22 26 29 29 32 2 Complex Numbers: Geometry 2.1 Geometric interpretation . . . . . . . 2.2 Circles and lines and geometric sets . 2.3 Moebius maps . . . . . . . . . . . . . 2.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 61 64 66

functions, which uses in the proof estimates on the logarithm function, and not the above estimates. Theorem 3.6.1 is still valid in a (possibly non-commutative) Banach algebra with identity, say B, with norm · such that ab ≤ a · b , but one loses a bit the speciﬁcity of complex numbers. The proof goes in the same way, with absolute value replaced by the norm of B. Taking B = CN ×N with the euclidean norm (or any other norm, since all norms are equivalent in a ﬁnite dimensional vector space) we

using real analysis might provide a student motivation to study complex analysis: ∞ rn sin(nθ) = n=1 r sin θ , 1 + r2 − 2r cos θ n−1 (z 2 − 2z cos( z 2n + 1 = k=0 cos t2 dt = R ∞ n=0 2n n 7n = π , 2 7 . 3 r ∈ [0, 1), 2k + 1 π) + 1), 2n θ ∈ R, 4 Prologue All these formulas are readily proved using complex analysis methods: The ﬁrst identity is easily proved by a purely real analysis method (multiply both sides by the denominator 1 + r2 − 2r cos θ, and use a trigonometric identity).

∞ p=1 But this is clear. tp p 1 1 − p−1 (n + 1)p−1 n ≤ 0. ≤ 0. 122 Chapter 3. Complex Numbers and Analysis Solution of Exercise 3.3.1. In view of (1.1.38) z 2n |z|2n , ≤ 2 + z n + z 5n 2(1 − |z|) the series with running term hand converges absolutely. |z|2n 2(1−|z|) z ∈ D, converges for |z| < 1 and hence the series at Solution of Exercise 3.3.2. For z = 0 the series trivially converges since every term is then equal to 0. Write 1 1 z + =− 2 . z−n n n (1 − nz ) Given any z ∈ C there