1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

10.4 • MAPPING BY TRIGONOMETR.IC FUNCTIONS 423

Therefore, we have

Arcsin {l + i) ·;::;: 0.666239432 + il.061275062.

Is there any reason to assume that there exists a conformal mapping for some
specified domain D onto another domain G? Our final theorem concerning the
existence of conformal mappings is attributed to Riemann and is presented in
Lars V. Ahlfors, Complex Analysis (New York: McGraw-Hill), Chapter 6 , 1966.

-------... EXERCISES FOR SECTION 10.4

1. Find the image of the semi-infinite strip - 4 w < x < 0, y > 0, under t he mapping
   w = tanz.


2. Find the image of the vertical strip 0 < Re(z) < j under the mapping w = tanz.


3. Find the image of the vertical line x = ~ under the transformation w = sin z.


4. Find the image of the horizontal line y = 1 under the transformation w =sin z.


5. Find t he image of the rectangle R = { x + iy: 0 < x < ~. 0 < y < 1} under the
   transformation w = sin z.


6. Find the image of the semi-infinite strip - 2 " < x < O, y > 0, under the mapping
   w = sinz.


7. (a} Find lim Arg [sin(t +iy}].
   v-+oo
   (b) Find lim Arg (sin( -;w + iy)J.
   11-+oo


8. Use Equations (10-26} and (10-27} to find
   (a) Arcsin (2 + 2i}.
   (b) Arcsin (- 2 + i).
   (c) Arcsin(l- 3i}.
   (d) Arcsin (- 4 -i).


9. Show that w = sinz maps t he rectangle R = {x+iy: 2" < x < !,O < y < b}
   one-to-one and onto the portion of the upper half-plane Im ( w} > 0 t hat lies inside
   the ellipse
   u2 v2
   --+--=!.
   cosh^2 b sinh^2 b