A Complex Analysis Problem Book by Daniel Alpay

By Daniel Alpay

This can be a selection of workouts within the conception of analytic capabilities, with accomplished and specified recommendations. we want to introduce the scholar to functions and elements of the speculation of analytic features now not continually touched upon in a primary direction. utilizing applicable routines we want to convey to the scholars a few elements of what lies past a primary direction in complicated variables. We additionally speak about issues of curiosity for electric engineering scholars (for example, the belief of rational services and its connections to the speculation of linear platforms and nation area representations of such systems). Examples of vital Hilbert areas of analytic features (in specific the Hardy house and the Fock house) are given. The ebook additionally encompasses a half the place proper evidence from topology, sensible research and Lebesgue integration are reviewed.

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6. Solutions 45 = = 1 1− 1− (1 − )(1 − ) π π (1 − cos ) + i sin 1− 11 11 . = = π 2(1 − Re ) 2(1 − cos ) 11 Hence 1 C= 2 π 1 π 11 and S = π = 2 cot 22 . 2. We set Cn to be the sum to be computed and n−1 Sn = sin(a + (2k + 1)b). k=0 Since b = 0 (mod π) we have that e2ib = 1 and we can write n−1 ei(a+(2k+1)b) Cn + iSn = k=0 n−1 = ei(a+b) (e2ib )k k=0 1 − e2ibn = ei(a+b) 1 − e2ib eibn e−ibn − eibn = ei(a+b) ib e e−ib − eib eibn sin bn = ei(a+b) ib e sin b i(a+nb) sin bn . =e sin b Thus taking real and imaginary parts on both sides, we get the required formula for Cn and also Sn = sin(a + bn) sin(bn) .

When do the expressions make sense? One defines the trigonometric functions and the hyperbolic functions for every complex number in terms of the exponential function as follows: eiz + e−iz , 2 eiz − e−iz , sin z = 2i ez + e−z , cosh z = 2 ez − e−z . 12) sinh(iz) = i sin z and cosh(iz) = cos z. 13) and similarly, All polynomial identities involving the trigonometric functions and the hyperbolic functions proved in calculus on the real line still hold in the complex plane. 2, which is not yet available at this stage of the book.

4) What happens for complex values of a? It will follow from the fundamental theorem of algebra that any non-constant polynomial with real coefficients can be factored as a product of terms of the form (z − r) with r ∈ R, and factors of the form (z − z0 )(z − z0 ) = z 2 − 2zRe z0 + |z0 |2 , z0 ∈ C \ R. 5. Polynomials 31 These last factors are called irreducible. They cannot be factored as products of polynomials with real coefficient. The following exercise illustrates these facts for some important polynomials.

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