Page 425 - Schaum's Outline of Theory and Problems of Advanced Calculus

416 FUNCTIONS OF A COMPLEX VARIABLE [CHAP. 16

Replacing x by x in the first integral and combining with the third integral, we find,
e e
ð R ix ix ð e iz ð e iz
dz ¼ 0
dz þ
dx þ
r x z z
HJA BDEFG
or
ð R sin x ð e iz ð e ix
2i dx ¼ dz dz
r x z z
HJA BDEFG
Let r ! 0 and R !1.ByProblem 16.33, the second integral on the right approaches zero. The first
integral on the right approaches
i
ð 0 ire ð 0
e
i
lim ire d ¼ lim ie ire i d ¼ i
r!0 re i r!0
since the limit can be taken under the integral sign.
Then we have
ð R sin x ð 1 sin x
lim 2i dx ¼ i or dx ¼
R!1 r x 0 x 2
r!0

MISCELLANEOUS PROBLEMS
2
16.36. Let w ¼ z define a transformation from the z plane (xy plane) to the w plane ðuv plane).
Consider a triangle in the z plane with vertices at Að2; 1Þ; Bð4; 1Þ; Cð4; 3Þ. (a) Show that the
image or mapping of this triangle is a curvilinear triangle in the uv plane. (b) Find the angles of
this curvilinear triangle and compare with those of the original triangle.
2
2
2
(a)Since w ¼ z ,we have u ¼ x y ; v ¼ 2xy as the transformation equations. Then point Að2; 1Þ in the
xy plane maps into point A ð3; 4Þ of the uv plane (see figures below). Similarly, points B and C map
0
into points B and C respectively. The line segments AC; BC; AB of triangle ABC map respectively
0
0
into parabolic segments A C ; B C ; A B of curvilinear triangle A B C with equations as shown in
0
0
0
0
0
0
0
0
0
Figures 16-12(a) and (b).

Fig. 16-12