csch {x - iy) 



sinh X cos y + i cosh x sin y 



sinh^ x + sin^ y 



sinh^ X sin'^ y + cosh^ x cos^ y = sinh^ x + cos^ y 



= cosh^ a; - sin^ V =— (cosh 2 ar + cos 2y) 



sinh-^ a; cos^ y + cosh a; sin^ y = sinh^ a; + sin^ y 

 = cosh'^ X - cos^ y =— (cosh 2 a; - cos 2 y) 



[32a] 



[32b] 



The first six of these formulas may serve as definitions; the others can be deduced from 

 them, or, for functions containing z, from results obtained in Section 20. In some cases deno- 

 minators are rationalized. In the last two formulas, the second and third members may be added 

 and divided by 2 in order to obtain the fourth. 



The first four functions are plotted in Figure 39. 



\ 















/ 



\ 















n 



> 









\ 



i<3, 



% 





\^ 











// 















J*/- 



C 

















--^ 



\ii 











/ 















/ 











lat 



|\_s^^ 



/ 















7 

















/ 



\ 













Y 





) 



i% 











1 







\ 











Figure 39 - Plots of four hyperbolic functions. 



68 



