vereinfachen-((e^{ipix}-e^{-ipix})^2)/4

Lösung

vereinfachen

- \frac{( e ^{iπ x} - e ^{- iπ x} ) ^{2}}{4}

sin^{2} (π x)

Schritte zur Lösung

- \frac{( e ^{iπ x} - e ^{- iπ x} ) ^{2}}{4}

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

= - \frac{( cos ( π x ) + i sin ( π x ) - ( cos ( - π x ) + i sin ( - π x ) ) ) ^{2}}{4}

(cos (π x) + sin (π x) i - (cos (- π x) + sin (- π x) i))^{2} = 2^{2} i^{2} sin^{2} (π x)

= - \frac{2 ^{2} i ^{2} sin ^{2} ( π x )}{4}

Streiche

\frac{2 ^{2} i ^{2} sin ^{2} ( π x )}{4} : i^{2} sin^{2} (π x)

= - i^{2} sin^{2} (π x)

i^{2} = - 1

= - (- 1) sin^{2} (π x)

Fasse zusammen

= sin^{2} (π x)

e x p an d \frac{25 ( x - 1 )}{3}

e x p an d \frac{1}{sin ( θ ) ( 1 + 2 cos ( θ ) )}

e x p an d (e^{2} - 2 ey + y^{2}) (e^{2} - 2 x + x^{2}) - (e^{2} - 2 x + x^{2}) r

e x p an d \frac{4 n ^{3} + 15 n ^{2} + 11 n}{6}

e x p an d (1 - x) y^{''} (x) + x y