vereinfachen e^{-iwt}(e^{ikz}+re^{-ikz})

Lösung

vereinfachen

e^{- i wt} (e^{ik z} + r e^{- ik z})

(cos (wt) cos (k z) + r cos (wt) cos (k z) + sin (wt) sin (k z) - r sin (wt) sin (k z)) + i (cos (wt) sin (k z) - cos (k z) sin (wt) - r cos (wt) sin (k z) - r cos (k z) sin (wt))

Schritte zur Lösung

e^{- i wt} (e^{ik z} + r e^{- ik z})

e^{- i wt} = cos (wt) - i sin (wt)

= (cos (wt) - i sin (wt)) (e^{- ik z} r + e^{ik z})

Wende imaginäre Zahlenregel an:

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

= (cos (wt) - i sin (wt)) (e^{- ik z} r + cos (k z) + i sin (k z))

e^{- ik z} = cos (k z) - i sin (k z)

= (cos (wt) - i sin (wt)) (r (cos (k z) - i sin (k z)) + cos (k z) + i sin (k z))

Wende arithmetische Regel für komplexe Zahlen an:

(a + bi) (c + d i) = (a c - b d) + (a d + b c) i

a = cos (wt), b = - sin (wt), c = cos (k z) + r cos (k z), d = sin (k z) - r sin (k z)

= (cos (wt) (cos (k z) + r cos (k z)) - (- sin (wt)) (sin (k z) - r sin (k z))) + (cos (wt) (sin (k z) - r sin (k z)) + (- sin (wt)) (cos (k z) + r cos (k z))) i

Vereinfache

cos (wt) (cos (k z) + r cos (k z)) - (- sin (wt)) (sin (k z) - r sin (k z)) : cos (wt) (cos (k z) + r cos (k z)) + sin (wt) (sin (k z) - r sin (k z))

Vereinfache

cos (wt) (sin (k z) - r sin (k z)) + (- sin (wt)) (cos (k z) + r cos (k z)) : cos (wt) (sin (k z) - r sin (k z)) - sin (wt) (cos (k z) + r cos (k z))

= (cos (wt) (cos (k z) + r cos (k z)) + sin (wt) (sin (k z) - r sin (k z))) + (cos (wt) (sin (k z) - r sin (k z)) - sin (wt) (cos (k z) + r cos (k z))) i

Entferne die Klammern:

(a) = a

= cos (wt) (cos (k z) + r cos (k z)) + sin (wt) (sin (k z) - r sin (k z)) + (cos (wt) (sin (k z) - r sin (k z)) - sin (wt) (cos (k z) + r cos (k z))) i

Schreibe

cos (wt) (cos (k z) + r cos (k z)) + sin (wt) (sin (k z) - r sin (k z)) + (cos (wt) (sin (k z) - r sin (k z)) - sin (wt) (cos (k z) + r cos (k z))) i

in der Standard komplexen Form um:

(cos (wt) cos (k z) + r cos (wt) cos (k z) + sin (wt) sin (k z) - r sin (wt) sin (k z)) + (cos (wt) sin (k z) - r cos (wt) sin (k z) - sin (wt) cos (k z) - r sin (wt) cos (k z)) i

= (cos (wt) cos (k z) + r cos (wt) cos (k z) + sin (wt) sin (k z) - r sin (wt) sin (k z)) + (cos (wt) sin (k z) - r cos (wt) sin (k z) - sin (wt) cos (k z) - r sin (wt) cos (k z)) i

e x p an d d (28.26)

e x p an d e^{2 t} (e^{t} - 1)

e x p an d d x ((x^{2} - 3 x + 2) \cdot (2 x^{3} + 1))

e x p an d \frac{k}{( s - 3 ) ( s + 11 )}

\frac{\partial}{\partial x} ((y) (3 x^{2} + y^{2} x + y))