The Remarkable Sine Functions

addition theorem algebraic analogous assume axis begin called Chapter circle circular functions coincides complex complex number complex variable condition Consequently consider coordinate corresponding cosh cosine cost curve defined definition denote derive differ elliptic functions entire equal equation example expressed fact follows formula 89 formulas fraction function inverse function sn geometric given hyperbolic functions hyperbolic sine imaginary increases integral interval Jacobian functions k² sn² lemniscate lemniscate functions lemniscate sine length m²² multiple namely nz¹ obtain origin parallelogram particular periodic function poles polynomial positive possible regard replace represented roots satisfied shown sine single sinh sint sl² sn² solution strips symmetric t₁ t₂ tion values vanishes yields zeros