Page 411 - Schaum's Outline of Theory and Problems of Advanced Calculus

402 FUNCTIONS OF A COMPLEX VARIABLE [CHAP. 16

@u @v @v @u
so that we must have and :
@x ¼ @y @x ¼ @y
Conversely, we can prove that if the first partial derivatives of u and v with respect to x and y are
continuous in a region, then the Cauchy–Riemann equations provide sufficient conditions for f ðzÞ to be
analytic.

16.8. (a)If f ðzÞ¼ uðx; yÞþ ivðx; yÞ is analytic in a region r, prove that the one parameter families of
2
curves uðx; yÞ¼ C 1 and vðx; yÞ¼ C 2 are orthogonal families. (b) Illustrate by using f ðzÞ¼ z .
(a) Consider any two particular members of these families uðx; yÞ¼ u 0 ; vðx; yÞ¼ v 0 which intersect at the
point ðx 0 ; y 0 Þ.
dy u x
Since du ¼ u x dx þ u y dy ¼ 0, we have :
dx ¼ u y
dy v x
Also since dv ¼ v x dx þ v y dy ¼ 0; ¼ : y
dx v y
When evaluated at ðx 0 ; y 0 Þ,these represent
respectively the slopes of the two curves at this
point of intersection.
By the Cauchy–Riemann equations, u x ¼
v y ; u y ¼ v x ,we have the product of the slopes at
the point ðx 0 ; y 0 Þ equal to
x

u x v x
¼ 1

u y v y
so that any two members of the respective families
are orthogonal, and thus the two families are ortho-
gonal.
2
2
2
(b)If f ðzÞ¼ z ,then u ¼ x y ; v ¼ 2xy. The graphs
2
2
of several members of x y ¼ C 1 ,2xy ¼ C 2 are Fig. 16-4
shown in Fig. 16-4.
16.9. In aerodynamics and fluid mechanics, the functions
and in f ðzÞ¼ þ i , where f ðzÞ is analytic, are called the velocity potential and stream
2
2
function, respectively. If ¼ x þ 4x y þ 2y,(a) find and (b) find f ðzÞ.
@ @ @ @
(a)Bythe Cauchy-Riemann equations, ¼ ; ¼ . Then
@x @y @x @y
@ @
¼ 2x þ 4 ¼ 2y 2
@y @x
ð1Þ ð2Þ
Method 1. Integrating (1), ¼ 2xy þ 4y þ FðxÞ.
Integrating (2), ¼ 2xy 2x þ Gð yÞ.
These are identical if FðxÞ¼ 2x þ c; Gð yÞ¼ 4y þ c, where c is a real constant. Thus,
¼ 2xy þ 4y 2x þ c.
Method 2. Integrating (1), ¼ 2xy þ 4y þ FðxÞ. Then substituting in (2), 2y þ F ðxÞ¼ 2y 2or
0
F ðxÞ¼ 2 and FðxÞ¼ 2x þ c. Hence, ¼ 2xy þ 4y 2x þ c.
0
2 2
ðaÞ From ðaÞ; f ðzÞ¼ þ i ¼ x þ 4x y þ 2y þ ið2xy þ 4y 2x þ cÞ
2 2 2
¼ðx y þ 2ixyÞþ 4ðx þ iyÞ 2iðx þ iyÞþ ic ¼ z þ 4z 2iz þ c 1
where c 1 is a pure imaginary constant.