| z/(z+i)|=4

Lösung

\frac{z}{z + i} = 4

z = x + \frac{- 16 + - 225 x ^{2} + 16}{15} i, z = x - \frac{- 225 x ^{2} + 16 + 16}{15} i; x \in R

Schritte zur Lösung

\frac{z}{z + i} = 4

Ersetze

z = x + y i

\frac{x + y i}{x + y i + i} = 4

Schreibe

\frac{x + y i}{x + y i + i}

um:

\frac{x ^{2} + y ^{2}}{x ^{2} + y ^{2} + 2 y + 1}

\frac{x ^{2} + y ^{2}}{x ^{2} + y ^{2} + 2 y + 1} = 4

Löse nach

y : y = \frac{- 16 + - 225 x ^{2} + 16}{15}, y = - \frac{- 225 x ^{2} + 16 + 16}{15}

Setze in

z = x + y i

ein

z = x + \frac{- 16 + - 225 x ^{2} + 16}{15} i, z = x - \frac{- 225 x ^{2} + 16 + 16}{15} i; x \in R

i (x - i y) = - i (x - i y)

(1 + i) z^{2} + (2 - i) z + (1 + 3 i) = 0

z^{2} = 49 + i

z^{2} + (2 i - 3) z + 5 - 1 = 0

s i f (x) = \frac{5 x ^{2}}{x}, f (0)