Complex Cosine Function/Examples/4 cos z = 3+i
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Example of Complex Cosine Function
Let:
- $4 \cos z = 3 + i$
Then:
- $z = \dfrac {\paren {8 n + 1} \pi} 4 - \dfrac {i \ln 2} 2$ for $n \in \Z$
or:
- $z = \dfrac {\paren {8 m - 1} i \pi} 4 + \dfrac {i \ln 2} 2$ for $m \in \Z$
Proof
\(\ds 4 \cos z\) | \(=\) | \(\ds 3 + i\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 4 \paren {\dfrac {e^{i z} + e^{-i z} } 2}\) | \(=\) | \(\ds 3 + i\) | Euler's Cosine Identity | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 e^{2 i z} - \paren {3 + i} e^{i z} + 2\) | \(=\) | \(\ds 0\) | mulitplying by $e^{i z}$ and rearranging | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds e^{i z}\) | \(=\) | \(\ds \dfrac {3 + i \pm \sqrt {\paren {3 + i}^2 - 4 \times 2 \times 2} } {2 \times 2}\) | Quadratic Formula | ||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {3 + i \pm \sqrt {9 - 1 + 6 i - 16} } 4\) | Square of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {3 + i \pm \sqrt {-8 + 6 i} } 4\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {3 + i \pm \paren {1 + 3 i} } 4\) | Square Root of $-8 + 6 i$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {4 + 4 i} 4 \text { or } \dfrac {2 - 2 i} 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + i \text { or } \dfrac 1 2 - \dfrac i 2\) |
Thus we have:
\(\ds e^{i z}\) | \(=\) | \(\ds 1 + i\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds i z\) | \(=\) | \(\ds \ln \paren {1 + i}\) | Definition 2 of Complex Logarithm | ||||||||||
\(\ds \) | \(=\) | \(\ds \ln \paren {\sqrt 2 \exp \paren {\dfrac {i \pi} 4} }\) | Definition of Exponential Form of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \sqrt 2 + \dfrac {i \pi} 4 + 2 n i \pi\) | Definition 1 of Complex Logarithm | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\ln 2} 2 + \dfrac {\paren {8 n + 1} i \pi} 4\) | simplification | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds -\dfrac {i \ln 2} 2 + \dfrac {\paren {8 n + 1} \pi} 4\) | multiplying through by $-i = \dfrac 1 i$ |
and:
\(\ds e^{i z}\) | \(=\) | \(\ds \dfrac 1 2 - \dfrac i 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds i z\) | \(=\) | \(\ds \ln \paren {\dfrac 1 2 - \dfrac i 2}\) | Definition 2 of Complex Logarithm | ||||||||||
\(\ds \) | \(=\) | \(\ds \ln \paren {\dfrac 2 {\sqrt 2} \exp \paren {-\dfrac \pi 4} }\) | Definition of Exponential Form of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \dfrac 2 {\sqrt 2} + \dfrac {-i \pi} 4 + 2 m i \pi\) | Definition 1 of Complex Logarithm | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac {\ln 2} 2 + \dfrac {\paren {8 m - 1} i \pi} 4\) | simplification | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds \dfrac {i \ln 2} 2 + \dfrac {\paren {8 m - 1} \pi} 4\) | multiplying through by $-i = \dfrac 1 i$ |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.6$. The Logarithm: Examples: $\text {(iii)}$