76 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



3.42 

 i. 



Cbsh- 1 /*+-)- sinh- 1 -(x - -Y 



2\ X/ 2\ Xj' 



= tanh- 1 = 2 tanh- 1 * 



x* + i x + i' 



= log x. 



2. cosh" 1 csc 2x = - sinh- 1 cot 2x = - tanh" 1 cos 2x y 



= log tan x. 



3. tanh- 1 tan 2 (- + -)= i log csc x. 



\4 2 / ! 



4. tanh" 1 tan 2 - = - log sec x. 



3.43 The Gudermannian. 



If, 



1. cosh x = sec 6. 



2. sinh x = tan 6. 



3. e x = sec + tan = tan 



f- + -V 



4. * = log tan + - 



5- B = gd x. 



3.44 



1. sinh x = tan gd x. 



2. cosh x = sec gd x. 



3. tanh x = sin gd x. 



4- tanh - = tan - gd x- 



2 2 



x _ i + sin gd x 

 cos gd x 



in f ^ + gd x\ 



sn 



