178 PIERCE. 



7. To Find cosh-i (P + jU), where P and U are Positive and 

 Real. — 



Let 



cosh-HP -\-jU) = A -{-jB, (22) 



then 



P + jU = cosh A cos B -\- j sinh A sin jB, 



whence 



P = cosh /I cos B, (23) 



[7 = sinh A sin 5. (24) 



The sum of the squares of these two equations gives 



P2 -j- u^= cosh2 J cos2 B + sinh2 .4 sin- B 

 = 1 + sinh- A — sin^ B, 



whence by substitution from (24) and by solution of the resulting 

 quadratic equations we obtain 



sinhM = - r ±Vr2+ V\ and 

 sin2 5= r±VL^2_|_pr2, 



These give, with choices of signs to make A and B real and satisfy 

 (24) 



if V >0, sinh A = ^V2Vg(h), sin B = ^V2V f{h), h = U/V, 



if F<0, sinh .4 = ±V'-2r/(/0,sin5 = ^V~^2V g{h), h = U/V. 



In accordance with (23) and (24), in each line the sinh A and sin B 

 have the same sign before their radicals, and the angle B must be so 

 determined that cos B is positive. Whence 



cosh-i (P + jU) = ± {a + j {cp + 27rn), (25) 



where 

 if F>0, a = sinh-M+V2r^(/0}, <p = sin-M+v'2r7(/0}, (26) 



if V<0, a = sinh-M+V-2F/(/0}, ^ = sin-H -\- V -2V g(h)} , (27) 



with h = U/V. (28) 



Equation {25) gives the value of cosh~i (P + jU), where P and U are 

 real, positive quantities, in terms of a and (p defined by {26), {27) and {28). 

 (p is in the first quadrani. V is defined by {14)- 



