Comments on Solutions

Page 4

The solution to Problem 4 is missing an integral sign at the beginning of the second line of the displayed formula, which should say \[\int_{a}^{b} f(\gamma(s)) \gamma'(s)\,ds.\]

In the solution to Problem 5, part (b), the notation \( [ a, (1-\delta)i,\infty)\) has not been defined. The meaning is the half-line that starts at the point \(a\) and passes through the point \( (1-\delta)i\), that is, \( \{ a+t( (1-\delta)i-a): t\ge 0\}\). The same notation appears elsewhere in the book, for instance at the top of page 2 in Chapter 3 and in Problem 3 on page 19 of Chapter 3.

Page 6

The solution to Problem 9 part (b) assumes knowledge of the complex sine function (mentioned in part (c)). Given that knowledge, one can shorten the solution by observing that the series in part (b) is the imaginary part of the series in part (a) when \(z\) is replaced by \(\exp(ix)\).

Page 8

In the solution to Problem 8, line 4, \(K_0\) should be \(K^\circ\). Throughout the solution, the symbol for the interior of \(K\) should be \(K^\circ\), not \(K^0\). (The notation is correct in the statement of the problem.)

Page 18

In the solution to Problem 7, the symbol \(i\) is used unfortunately both as a summation index and as \(\sqrt{-1}\).

Page 19

In the second line, insert an opening left-hand bracket following the first equals sign.

Page 20

In the solution to Problem 9(d), second line, insert a closing right-hand parenthesis preceding the word “Let”.

In the solution to Problem 10(d), first line, a fraction bar is missing in the expression at the end of the line, which should be \( \exp[\sin(1/z)/\cos(1/z)]\).

Page 20

In the second line of the display at the bottom of the page, delete the period following \(|a|\).

Page 21

In the first displayed formula, the restriction that “\(z\in U\)” is not needed, for the point \(z\) is automatically in the set \(U\) when \(|z|\) exceeds the maximum of \(|a|\) and \(|b|\).

In the second displayed formula, there should be a factor of \(z^n\) in the sum.

In order for the question of a Laurent series of \(g\) in the region where \(|z|\lt \max(|a|,|b|)\) to make sense, there needs to be an assumption that \(0\) is not a point of the line segment \([a,b]\).

Harold P. Boas