Question

Find the real part of $z={{e}^{{{e}^{i\theta }}}}$, $\theta \in R$.

Answer 1

- Hint: Use Euler’s identity and find the value of ${{e}^{i\theta }}$,which is of the form. Thus after getting the expression raise the value of ${{e}^{i\theta }}$ to e. Split the expression into real and imaginary parts and apply Euler's identity in the imaginary part. Thus simplify the expression obtained and get the real part of the expression.

Complete step-by-step solution -

We have been asked to find the real part of the given expression, $z={{e}^{{{e}^{i\theta }}}}$. We know that, $i=\sqrt{-1}$.
A complex number is represented as $a+ib$, where a is the real part and b is the imaginary part.
Now we know that ${{e}^{i\theta }}$ is of the form $\left( \cos \theta +i\sin \theta \right)$.
i.e. ${{e}^{i\theta }}=\cos \theta +i\sin \theta -(1)$
Now this expression is known as Euler’s formula. It is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.
Here e is the base of the natural logarithm, i is the imaginary unit, and $\cos$ and $\sin$are the trigonometric functions.
Here, ${{e}^{{{e}^{^{i\theta }}}}}={{e}^{\cos \theta +i\sin \theta }}$, from (1)
Now let us split up and write it as,
${{e}^{\cos \theta +i\sin \theta }}={{e}^{\cos \theta }}.{{e}^{i\sin \theta }}$
Now in the ${{2}^{nd}}$ term ${{e}^{i\sin \theta }}$, we can again expand it as ${{e}^{i\theta }}$ put, $\theta =\sin \theta$.

& {{e}^{\cos \theta }}.{{e}^{i\sin \theta }}={{e}^{\cos \theta }}\left[ \cos \left( \sin \theta \right)+i\sin \left( \sin \theta \right) \right] \\\ & {{e}^{\cos \theta }}.{{e}^{i\sin \theta }}={{e}^{\cos \theta }}.\cos \left( \sin \theta \right)+{{e}^{\cos \theta }}\left[ i\sin \left( \sin \theta \right) \right] \\\ \end{aligned}$$ From the above expression, the real part and imaginary part is of the form $$a+ib$$. $$a+ib={{e}^{\cos \theta }}.\cos \left( \sin \theta \right)+{{e}^{\cos \theta }}\sin \left( \sin \theta \right)i$$ Thus real part $$=a={{e}^{\cos \theta }}\cos \left( \sin \theta \right)$$ Imaginary part $$=b={{e}^{\cos \theta }}\sin \left( \sin \theta \right)$$ Note: Here by Euler’s identity we got the value of $${{e}^{i\theta }}$$. Euler’s identity is important in mathematics because it combines five constants of maths and three mathematical operations, each occurring one at a time. The three operations that it contains are exponentiation, multiplication and addition. The five constants are numbers 0, 1, $$\pi $$, e and i. Thus we have a standard value that we can use directly.