SH

y=shx是初等函數中的雙曲正弦函數 shx=(e^x-e^-x)/2

1、定義域：R

2.值域：（e^x+e^-x)/2≥2/2（√e^x*e^-x)=1

3.奇偶性：sh(-x)=[e^-x-e^-(-x)]/2=-shx，奇函數.<----

shx=(e^x-e^-x)/2

sh(-x)=(e^-x-e^x)/2=-shx

因此函數shx為奇函數

sh(x+y)=shxchy+chxshy

證：shxchy+chxshy ^[1] 

=[(e^x-e^-x)(e^y+e^-y)+(e^x+e^-x）（e^y-e^-y)]/（2*2）

=(e^x*e^y - e^-x*e^y + e^x*e^-y - e^-x*e^-y + e^x*e^y + e^-x*e^y - e^x*e^-y - e^-x*e^-y)/4

=（2e^x*e^y - 2e^-x*e^-y)/4

=[e^(x+y)-e^-(x+y)]/2

=sh(x+y)

sh(x-y)=shxchy-chxshy

證：sh(x-y)

=sh[x+(-y)]

=shxch(-y)+chxsh(-y)

=shxchy-chxshy