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Math Help - Questions from 'Fudamentals of Complex Analysis' by Saff

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    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
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    Quote Originally Posted by Aileys. View Post
    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 \arg(x+iy) can be computed using the form

    \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 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 sgn(y)\arccos \frac{x}{\sqrt{x^2+y^2}} , at its points of continuity, equals \arg(x+iy)

    16) Prove that ||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) (1-\sqrt{3}i)^{1/3} ; (f) \left(\frac{2i}{1+i}\right)^{1/6}

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

    10) Find all four roots of the equation z^4+1=0 and use them to deduce the factorization 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
    .
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