which is transformed into the upper half of the ^-plane and hence may appropriately be defined 

 to constitute the interior. An infinite polygon can be traced in either direction; if the direction 



is reversed, the exterior angles are replaced by their supplements, and the former exterior be- 

 becomes the interior. 



Detailed discussions of a number of cases will be found in the next chapter. 



32. THE HYPERBOLIC FUNCTIONS 



The following formulas are collected here for convenience of reference. Where - occurs 



twice in the same formula, the upper sign is to be taken throughout the formula, or the lower 



sign throughout. The positive square root is always meant, and In denotes the logarithm to 



base e. 



sinh a; =— (e*-e~*), cosh a; =-l-(e^+ e~ ^ 

 2 2 



tanhcr=liH!L£=l!z^ , coth x = "-^^^^ = e^^ + eZl 

 cosh a; „x.„-x sinh x -x -x 



sech X = —± — = '^ _ , csch x «^- 



2 



cosh a; e* + e~^' sinh x e*-e~* 



sinh (a!±y) = sinh x cosh y ± cosh x sinh y 

 cosh (x-y) - cosh x cosh y - sinh x sinh y 

 sinh 2a; = 2sinh x cosh x, cosh 2a? = cosh^ x + sinh^ x 



tanh 2a! - ^ ^^"^ ^ . coth 2a; = ^ (tanh x + coth a;) 

 l + tanh2ar 2 



sinhia! = i \J±-(cosh. x - 1) cosh— a; = J— (cosh a; + l) 

 2 2 2 2 



(The sign is + or - according as the value of x is + or - ) 



X = 



coth I X = «i"') ^ 



2 cosh x + 1 2 cosh ar-1 



sinh-l a; = In (x + Jx'^ + 1), tanh" ^ a; =i. In i±^ 



^ ^ '^ 2 1-ar 



cosh~l a! = ±ln (x + x/^^-l), coth" ^ a; =1 ln-^±i 



2 x-1 



— sinh X = cosh x — cosh x = sinh x 



dx .dx 



