10.3 • MAPPINGS INVOLVING ELEMENTARY FUNCTIONS 417(a) (b)
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Figure 10. 15 The Riemann surfaces for the mapping w = (z^2 - 1)'.
-------... EXERCISES FOR SECTION 10.3
- Find the image of the semi-infinite strip 0 < x < ~, y > 0, under t he transforma-
tion w =exp (iz). - Find the image of the rectangle 0 < x < In 2, 0 < y < ~, under the transformation
w = expz. - Find the image of the first quadrant x > 0, y > 0, under w = ~L-Ogz.
- Find the image of t he annulus 1 < lzl < e under w = Logz.
- Show that t he multivalued function w =log z maps t he annulus 1 < lzl < e onto
the vertical strip 0 < Re ( w) < 1. - Show that w =^2 ; ;
2
maps t he portion of the right half-plane Re (z) > 0 that lies
to the right of the hyperbola x^2 - y^2 = 1 onto t he unit disk lwl < 1. - Show that the function w = ::+! maps the horizontal strip -71' <Im (z) < 0 onto
the region 1 < lw l. - Show t hat w = ::+~ maps the horizontal strip IYI < f onto t he unit disk lwl < 1.
- Find the image of the upper half-plane Im (z) > 0 under w =Log~.
- Find the image of t he portion of the upper half-plane I m (z) > 0 that lies outside
the circle lzl = 1 under the t ransformation w = Log ( ~).
- Show that t he function w = (1 + z)^2 / (1 - z)^2 maps the portion of the disk lz l < 1
that lies in the first quadrant onto the portion of t he upper half-plane Im (w) > 0
t hat lies outside t he unit disk. - Find the image of the upper half-plane Im ( z) > 0 under w = Log ( 1 - z^2 ).
1 3. Find t he branch of w = ( z^2 + 1) ~ that maps the right half-plane Re ( z) > 0 onto
the right half-plane Re(w) > 0 sHt along the segment 0 < u $ 1, v = 0.
2- Show t hat the transformation w = ; 2 +: maps the portion of the first quadrant
x > 0, y > 0, that lies outside t he citcle lzl = 1 onto the first quadrant u > 0,
v > 0.