44] 



FUNCTIONS OF A COMPLEX VARIABLE. 



87 



The curves 



FIG. 22. 



= const, and v = 



,.= const., 

 = 0, 



x> -H y 2 - Cjx = 0, # 2 + ?/ 2 + 



give two sets of circles, the first all tangent to the F-axis at 

 the origin, the second all tangent to the X-axis, Fig. 23. 



The power 



z n = (x + iy) n = r n {cos n<t> + i sin ??$), 

 gives the two functions 



u r n cog n( v 



and a sum of any number of such terms each multiplied by a con- 

 stant 



2r w { A n cos n<j) + B n sin nty, 



therefore gives a harmonic function. If a function can be developed 

 in such a trigonometric series it accordingly is harmonic. Terms 

 such as these may be called circular harmonic functions. 



