Which of the following is equal to $e^{ar{z}}$, where $z = x + iy$ with $x$ and $y$ real numbers?
A - HSC - SSCE Mathematics Extension 2 - Question 8 - 2024 - Paper 1
Question 8
Which of the following is equal to $e^{ar{z}}$, where $z = x + iy$ with $x$ and $y$ real numbers?
A. $\bar{e}$
B. $e^{-z}$
C. $e^{2x} e^{\bar{z}}$
D. $e^{-2z} e^{\... show full transcript
Worked Solution & Example Answer:Which of the following is equal to $e^{ar{z}}$, where $z = x + iy$ with $x$ and $y$ real numbers?
A - HSC - SSCE Mathematics Extension 2 - Question 8 - 2024 - Paper 1
Step 1
Identify the expression
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Answer
The expression we need to analyze is e^{ar{z}}. Since z=x+iy, we have ar{z} = x - iy. Thus, we want to find an equivalent expression for ex−iy.
Step 2
Apply Euler's formula
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Answer
Using Euler's formula, we know that ea+bi=ea(extcos(b)+iextsin(b)). Substituting a=x and b=−y, we get:
e^{ar{z}} = e^{x - iy} = e^x ( ext{cos}(-y) + i ext{sin}(-y)) = e^x ( ext{cos}(y) - i ext{sin}(y)).
Step 3
Compare with options
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Answer
Now, we can compare this result with the given options:
A. eˉ does not match.
B. e−z=e−x−iy=e−xe−iy. Hence, this is not equivalent either.
C. e2xezˉ=e2xex−iy=e3xe−iy, which is incorrect.
D. e−2zezˉ=e−2(x+iy)ex−iy=e−2xe−2iyex−iy. This simplifies to e−xe−3iy, which is also not valid.
After evaluating all the options, it is clear that the most suitable answer is A.
