Question

Which one of the following functions is analytic, given \( i = \sqrt{-1} \)?

• A. \( e^x (\cos y + i \sin y) \)
• B. \( e^x (\cos y - i \sin y) \)
• C. \( e^x (-\cos y + i \sin y) \)
• D. \( e^{-x} (\cos y + i \sin y) \)

Hint:

When checking if a function is analytic, you can use Euler's formula \( e^{i y} = \cos y + i \sin y \), which is analytic. Products of analytic functions are also analytic.

Answer 1

The Correct Option is A

A function is analytic if it satisfies the Cauchy-Riemann equations in the domain of interest. The function \( e^x (\cos y + i \sin y) \) is a product of two functions: \[ e^x \quad {and} \quad \cos y + i \sin y. \] The second function is a known Euler’s formula expression for \( e^{iy} \), which is analytic for all real values of \( y \). The exponential function \( e^x \) is also analytic for all real values of \( x \). Therefore, the product of these two functions is analytic, and the correct answer is option (A).