Expressing Solutions of Laplace's Equation. 195 



In the second integral write u = '2'jt — u! , 



j /,„ (p cos u + iz) sin ;?-??/ (/?i = 1 /,„ (p cos ?i + iz) sin mi/ c??/ 



+ 1 /m (p COS u' + i^) sin mu'du' 



= 0. 

 Hence 



r 



1 ///J ('^' cos v+y sin r + i^') cos mv dv 



^\ fm (p (ios u + iz) cos mudu (1) 



Jo 



COS 772 



But the f's and F^s being arbitrary functions, we may replace 

 fjn (p COS u + iz) by /,„ {z — ip cos u) . The general expression 

 for V then becomes 



V = | f^^{z — ipcQ>'iu)du-^ 



. 

 + cos ni^\ /",„ [z — ip cos %i\ cos 7?12* f??/ + 



Jo 



+ sin >>?</) I F;„(^ — ?/3cos2z) cos7)i?/^7f 4- (2) 



Jo 



3. Use now the symbolic form of Taylor"'s Theorem, 



d 

 — la eo9 u — 



/„(^-/pcos»)=« "-/„(.). ... (3) 



V may be written symbolically 



+ COS m0 M ( ^~ ''^ ''"^ " ^^ j cos ?>m du /m (■s') + 



+ sin m^ [ r^'^e " '■'' '°^ '^ ^^^ cos 7.71. d^ Yj^z) + (4) 



Bnt it is easy to show that 



V\-''^''^'''-QQ-<mudu^l{-\y\ e"'''''''cosmudu . (5) 



Jo Jo 



rV''^°'"cOS7772*(/7/. = 7ri'»J,J(^) (6) 



Jo 

 (See Gray & Matthews, ' Bessel Functions/ p. 89.) 

 2 



and 



