Complex problem in complex numbers

Algebra Level 5

For $i = \sqrt{-1}$ , let $z$ be a complex number satisfying $z = a^i$ . Denote $f(x)$ and $g(x)$ as the real part and imaginary part of $z$ respectively, that is $f(x) = \Re(z) , g(x) = \Im(z)$ .

Let the intersection point of $f(x)$ and $g(x)$ be expressed as $(m,n)$ in the coordinate axes.

Denote $m_n$ the $n^\text{th}$ absissa, where $m_n < 1$ . Compute $m_1 + m_2 + m_3 + \cdots$ .

e^{\sin 2} - \ln 2

\frac{e^\pi}{\pi - 1}

\frac{e^{\pi /4}}{e^\pi - 1}

\pi + e^\pi

1 solution

Subh Mandal
Apr 6, 2016

a^i=z iln(a)=ln(z) e^(iln(a))=z Cos(lna)+isin(lna)=z Fx=cos(lna) Gx=sin(lna) F(x)=g(x) Sin(lna)=sin(π/2-lna) Lna=nπ+(-1)^n(π/2-lna) Only even number n possible So n=-2,-4........ Form a gp of ratio e^(-π) with first term e^(-3π/4) Hence formula till infinity =f/(1-r) Hence option (e^π/4)/e^π-1 is correct

Complex problem in complex numbers

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