化简 x(e^{iy}+e^{i2piy})

解答

化简

x (e^{i y} + e^{i 2 π y})

(x cos (y) + x cos (2 π y)) + i (x sin (2 π y) + x sin (y))

求解步骤

x (e^{i y} + e^{i 2 π y})

使用虚数运算法则:

e^{ia} = cos (a) + i sin (a)

= x (e^{2 iπ y} + cos (y) + i sin (y))

使用虚数运算法则:

e^{ia} = cos (a) + i sin (a)

= x (cos (y) + i sin (y) + cos (2 π y) + i sin (2 π y))

将

x (cos (y) + sin (y) i + cos (2 π y) + sin (2 π y) i)

改写成标准复数形式:

(x cos (y) + x cos (2 π y)) + (x sin (y) + x sin (2 π y)) i

= (x cos (y) + x cos (2 π y)) + (x sin (y) + x sin (2 π y)) i

e x p an d \frac{1}{( s + a ) ( s + b ) ^{2}}

e x p an d \frac{3 ( x + 2 )}{5}

e x p an d (x - 3)^{3}

e x p an d \frac{1 - e ^{- 2 π s}}{( s ^{2} + 1 ) ( s ^{2} + 4 )}

e x p an d \frac{7 e ^{x}}{( 7 + e ^{x} ) ^{2}}