1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

3.3 • HARMONIC FUNCTIONS 119

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Bquiporential

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Streamline

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Figure 3.5 The families of orthogonal curves {(x,y): <f>(x,y) =constant} and

{(x, y) : ..P (x, y) = constant} for the function F (z) = 4> (x, y) + i,P (x, y).

y

The fluid flow V(x,y) = 2x -i 2y

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/' S!rea.mline

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Figure 3.6 T he equipotential cu.rves x^2 - y^2 = C and streamline curves 2xy = C for

the function F (z) = z^2 •

-------.. EXERCISES FOR SECTION 3.3

1. Determine where the foUowing functions are harmonic.

(a) u (x, y) = e% cosy and t1 (x , y) = e% sin y.

(b) u(x,y) = ln(x^2 +y^2 ) for (x,y) # (0,0).