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or ic — k = — 1, whence c = and & = 1, when e = 2.71 . . . The 

 functions (p and 4' are therefore 



,,U) =1-- + -, -•■• 



Had we begun with the construction a'^ where a is any positive constant 

 other than e we should have found k equal to loge a. 



11. The construction of e' = e'^e'^ is, on multiplying the series 



for all values for the complex variable z. 



The even and odd parts of e^ are named the hyperbolic cosine and sine 

 of z. The even and odd parts of e"' are named the circular cosine and 

 sine of z. In symbols written respectively 



cosh z = 1 + — +■■■ , sinh z = z + — + ■ ■ ■ , 



cos z = 1 — — +•■• , sin z = 2 — — I + • • • . 



In general for any integral number z 



e^ — cosh z + sinh z, 



e''= = cos z + i sin z. 

 Also 



cosh iz = cos 2, sinh iz = i sin z, 

 cos iz = cosh 2, sin iz = i sinh z. 



The theory of these functions can now be analytically developed. 



