三角函数
用幂级数展开可以得到Euler公式
%2Bi%5Csin(x).%0A#card=math&code=e%5E%7Bix%7D%3D%5Ccos%28x%29%2Bi%5Csin%28x%29.%0A&id=g56MD)
假设是实数，那么
%20%3D%20(1%2F2)(e%5E%7Bix%7D%2Be%5E%7B-ix%7D)%2C%20%5C%3B%20%5Csin(x)%20%3D%20(1%2F2i)(e%5E%7Bix%7D-e%5E%7B-ix%7D).#card=math&code=%5Ccos%28x%29%20%3D%20%281%2F2%29%28e%5E%7Bix%7D%2Be%5E%7B-ix%7D%29%2C%20%5C%3B%20%5Csin%28x%29%20%3D%20%281%2F2i%29%28e%5E%7Bix%7D-e%5E%7B-ix%7D%29.&id=ze53I)
双曲三角函数
现在定义:
双曲余弦函数
双曲正弦函数
和双曲正切函数
容易证明
- (双曲三角函数的勾股定理)
- (导函数)
- (角相加)
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由此得到二倍角公式
- (反函数) 如果
那么当时，有
应用
双曲三角函数代换在计算不定积分时有用，被称为Euler代换. 例如：
求原函数%5E2%7D#card=math&code=%5Cint%20%5Cfrac%7B%5Cmathrm%7Bd%7Dx%7D%7B%28x%2B%5Csqrt%7B1%2Bx%5E2%7D%29%5E2%7D&id=ktHdx).
解：设#card=math&code=x%20%3D%20%5Csinh%28t%29&id=t9M1B), 则
. 原积分变成
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