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Question Number 217101 by MathematicalUser2357 last updated on 01/Mar/25
is this right when (a+bi)^(c+di) =∣a+bi∣^(c+di) e^(i(c+di)arg(a+bi)) ? I had let arg(a+bi)= { ((tan^(−1) ((b/a))),(a≥0 and b≥0)),((π−tan^(−1) (−(b/a))),(a<0 and b≥0)),((−(π−tan^(−1) ((b/a)))),(a<0 and b<0)),((−tan^(−1) ((b/a))),(a≥0 and b<0)) :} before I solved it (a+bi)^(c+di) =∣a+bi∣^(c+di) e^(i(c+di)arg(a+bi)) =∣a+bi∣^c ∣a+bi∣^di e^(ic∙arg(a+bi)) e^(−d∙arg(a+bi)) =∣a+bi∣^c (c^di )^(ln∣a+bi∣) e^(ic∙arg(a+bi)) e^(−d∙arg(a+bi))
isthisrightwhen(a+bi)c+di=∣a+bi∣c+diei(c+di)arg(a+bi)?Ihadletarg(a+bi)={tan−1(ba)a⩾0andb⩾0π−tan−1(−ba)a<0andb⩾0−(π−tan−1(ba))a<0andb<0−tan−1(ba)a⩾0andb<0beforeIsolvedit(a+bi)c+di=∣a+bi∣c+diei(c+di)arg(a+bi)=∣a+bi∣c∣a+bi∣dieic⋅arg(a+bi)e−d⋅arg(a+bi)=∣a+bi∣c(cdi)ln∣a+bi∣eic⋅arg(a+bi)e−d⋅arg(a+bi)

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