Let $z = x + iy$ and $\arg\left(e^z\right) = \arg\left(e^{i(z+y)}\right)$. If $y = f(x)$ is a function, then $f(3)$ is equal to ____.
Step-by-Step Solution
Key Concept: The argument of $e^w$ equals the imaginary part of $w$ when expressed in the form $re^{i\theta}$.
Let $z = x + iy$. Then $e^z = e^{x+iy} = e^x e^{iy}$, so $\arg(e^z) = y$. For the right side, $e^{i(z+y)} = e^{i(x+iy+y)} = e^{i(x+2iy)} = e^{-2y} e^{ix}$, so $\arg(e^{i(z+y)}) = x$. Setting these equal: $y = x$, which gives $f(x) = x$. Therefore $f(3) = 3$.
Correct Answer: 0.20