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[공학수학] 17장 복소함수미분 삼각함수

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Trigometric and hyperbolic function

 

eix=cosx+isinxeix=cosxisinxcosx=eix+eix2sinx=eixeix2i

 

cosz=cosxcoshyisinxsinhycosz=eiz+eiz2=ei(x+yi)+ei(x+yi)212(ey(cosxisinx))+12(ey(cosx+isinx))12cosx(ey+ey)+12isinx(eyey

 

Similarlysinz=sinxcoshy+icosxsinhy

 

 

|cosz|2=cos2xcosh2y+sin2xsinh2y=cos2x(1+sinh2y)+sin2xsinh2y=cos2x+sinh2ySimlarily|sinz|2=sin2x+sinh2y

 

coshz=12(ez+ez)sinhz=12(ezez)coshiz=coszsinhiz=isinzcosiz=coshzsiniz=isinhz