Thread: Questions from 'Fudamentals of Complex Analysis' by Saff

1. Questions from 'Fudamentals of Complex Analysis' by Saff

Would some kind soul with access to this book pretty please post the following questions for me? I need to do these for my class and all the copies in the library are gone & all the ones from the bookshop sold out!

Section 1.3: questions 14 & 16
Section 1.5: question 5 parts d & f, question 8, question 10
Section 1.6: questions 11 & 12
Section 2.1: question 2 parts c & d, question 3 parts b & d, question 4,question 10 parts b & d

Thanks so much to anyone that can help me out

2. Originally Posted by Aileys.
Would some kind soul with access to this book pretty please post the following questions for me? I need to do these for my class and all the copies in the library are gone & all the ones from the bookshop sold out!

Section 1.3: questions 14 & 16

14) Show that a correct formula for $\displaystyle \arg(x+iy)$ can be computed using the form

$\displaystyle \arg(x+iy)=\left\{\begin{array}{ll}\arctan(y/x)+\frac{\pi}{2}[1-sgn(x)]&\,if\,\,x\neq 0\\\frac{\pi}{2}sgn(y)&\,if\,\,x=0\,,\,y\neq 0\\unde{f}ined &\,if\,\,x=y=0\end{array}\right.$

with $\displaystyle sgn(t)=\left\{\begin{array}{rl}1&\,if\,\,t>0\\0&\, if\,\,t=0\\-1&\,if\,\,t<0\end{array}\right.$

Show also that the expression $\displaystyle sgn(y)\arccos \frac{x}{\sqrt{x^2+y^2}}$ , at its points of continuity, equals $\displaystyle \arg(x+iy)$

16) Prove that $\displaystyle ||z_1|-|z_2||\leq |z_1-z_2|$

Section 1.5: question 5 parts d & f, question 8, question 10

5) Find the values of (d) $\displaystyle (1-\sqrt{3}i)^{1/3}$ ; (f) $\displaystyle \left(\frac{2i}{1+i}\right)^{1/6}$

8) Let $\displaystyle a,b,c\in\mathbb{R}\,,\,a\neq 0$ . Show that the equation $\displaystyle az^2+bz+c=0$ has (a)two real solutions if $\displaystyle b^2-4ac>0$ , (b) two non-real conjugate solutions if $\displaystyle b^2-4ac <0$

10) Find all four roots of the equation $\displaystyle z^4+1=0$ and use them to deduce the factorization $\displaystyle z^4+1=(z^2-\sqrt{2}z+1)(z^2+\sqrt{2}z+1)$

Section 1.6: questions 11 & 12
Section 2.1: question 2 parts c & d, question 3 parts b & d, question 4,question 10 parts b & d

Thanks so much to anyone that can help me out
.