smc-1191-65-Conjugates

Complex Numbers, EXT2 N1 2024 HSC 8 MC

Which of the following is equal to \(e^{\large{\bar{z}}}\), where \(z=x+i y\) with \(x\) and \(y\) real numbers?

\(\overline{e^{\large{z}}}\)
\(e^{\large{-z}}\)
\(e^{\large{2 x}} e^{\large{z}}\)
\(e^{\large{-2 x}} e^{\large{z}}\)

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\(A\)

Show Worked Solution

	\(e^{\large{\bar{z}}}\)	\(=e^{x-iy}\)
		\(=e^x \cdot e^{-iy}\)
		\(=e^x(\cos (-y)+i \sin (-y))\)
		\(=e^x(\cos (y)-i \sin (y))\)

	\(\overline{e^{\large{z}}}\)	\(=\overline{e^x(\cos (y)+i \sin (y)}\)
		\(=e^x(\cos (y)-i \sin (y))\)
		\(=e^{\large{\bar{z}}}\)

\(\Rightarrow A\)