integral of 1/(sqrt(2))e^{pixi(n-m)}

解

\int \frac{1}{2} e^{π x i (n - m)} d x

\frac{2}{2 π ( n - m )} sin (πn x - πm x) - i \frac{2}{2 π ( n - m )} cos (πn x - πm x) + C

解答ステップ

\int \frac{1}{2} e^{π x i (n - m)} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int e^{π x i (n - m)} d x

虚数の規則を適用する:

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

= \frac{1}{2} \cdot \int cos (πn x - πm x) + i sin (πn x - πm x) d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int cos (πn x - πm x) d x + \int i sin (πn x - πm x) d x)

\int cos (πn x - πm x) d x = \frac{1}{πn - πm} sin (πn x - πm x)

\int i sin (πn x - πm x) d x = - i \frac{1}{πn - πm} cos (πn x - πm x)

= \frac{1}{2} (\frac{1}{πn - πm} sin (πn x - πm x) - i \frac{1}{πn - πm} cos (πn x - πm x))

簡素化

\frac{1}{2} (\frac{1}{πn - πm} sin (πn x - πm x) - i \frac{1}{πn - πm} cos (πn x - πm x)) : \frac{2}{2 π ( n - m )} sin (πn x - πm x) - i \frac{2}{2 π ( n - m )} cos (πn x - πm x)

= \frac{2}{2 π ( n - m )} sin (πn x - πm x) - i \frac{2}{2 π ( n - m )} cos (πn x - πm x)

定数を解答に追加する

= \frac{2}{2 π ( n - m )} sin (πn x - πm x) - i \frac{2}{2 π ( n - m )} cos (πn x - πm x) + C