1550251515-Classical_Complex_Analysis__Gonzalez_

510 Chapter^7

y

0 x

Fig. 7.30

the integral

j Wr ds = J u dx + v dy (7.29-3)

c+ c+

exists and it is called the circulation around C. Similarly, the integral

J Wv ds = j u dy - v dx (7.29-4)

c+ c+

exists and it is called the flux through C.

If the integral (7.29-3) is zero for every C such that C* C D' C D, the

flow w = u +iv is said to be irrotational in D', and if the integral (7.29-4)

is zero for every C such that C* C D', the flow is said to be solenoidal

in D'. Suppose that u and v are continuously differentiable in D' and

that D' is a simply connected subdomain of D. Then as a consequence

of Green's theorem it follows that a given flow w = u +iv is irrotational

and solenoidal in D' if and only if -v(x, y) is the harmonic conjugate of

u(x, y) in D', i.e., such that Uy = v., and Ux = -vy for all (x, y) E D'.

Hence u -iv is then an analytic function in D', and w = u +iv is conjugate

analytic in the same region.

Let z 0 be an arbitrary (but fixed) point in D', z a variable point in the

same region, and 'Y any regular arc joining z 0 and z. Then the function

F'(z)=U+iV=('r) t wdz=(1) t(u-iv)dz

Jzo ~zo

(7.29-5)

is, except for an arbitrary constant depending on the choice of z 0 , a unique

single-valued analytic function in D', such that

F'(z) = u -.iv