To find a complex solution $z$ for the equation $e^{2z} = 2 + i$, let's break down the solution process into steps.

1. Express the Equation in Terms of Natural Logarithms: The equation is: $e^{2z} = 2 + i$ To isolate $z$, take the natural logarithm of both sides: $2z = \ln(2 + i)$ Then: $z = \frac{1}{2} \ln(2 + i)$

2. Convert $2 + i$ to Polar Form: To find $\ln(2 + i)$, express $2 + i$ in polar form as $re^{i\theta}$, where:

   • $r = |2 + i| = \sqrt{2^2 + 1^2} = \sqrt{5}$,
   • $\theta = \arg(2 + i) = \tan^{-1} \left(\frac{1}{2}\right)$.

   So, $2 + i = \sqrt{5} e^{i \tan^{-1}(1/2)}$

3. Calculate the Complex Logarithm: Using the polar form, we have: $\ln(2 + i) = \ln(\sqrt{5}) + i \tan^{-1}\left(\frac{1}{2}\right)$ which simplifies to: $\ln(2 + i) = \frac{1}{2} \ln(5) + i \tan^{-1}\left(\frac{1}{2}\right)$

4. Divide by 2 to Solve for $z$: Now, divide by 2: $z = \frac{1}{2} \left(\frac{1}{2} \ln(5) + i \tan^{-1}\left(\frac{1}{2}\right)\right)$ Simplifying further, we obtain: $z = \frac{\ln(5)}{4} + \frac{i}{2} \tan^{-1}\left(\frac{1}{2}\right)$

5. Account for Multiple Solutions: Since the logarithm of a complex number is multi-valued, the argument $\theta = \tan^{-1}(1/2)$ has an infinite number of values: $\theta = \tan^{-1}\left(\frac{1}{2}\right) + 2\pi k, \quad k \in \mathbb{Z}$ Each value of $k$ will yield a different solution for $z$: $z = \frac{\ln(5)}{4} + \frac{i}{2} \left(\tan^{-1}\left(\frac{1}{2}\right) + 2\pi k\right)$

This expression represents the general solution for $z$.

Would you like further details on any step?

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Questions for Further Exploration:

1. How can the polar form of a complex number be used to solve other equations?
2. What does it mean for a function to have multiple complex solutions?
3. How does the multi-valued nature of the logarithm affect complex analysis?
4. Why does dividing by 2 affect the argument $\theta$ in finding solutions?
5. What are some practical applications of complex solutions in physics or engineering?

Tip: Remember, the logarithm of a complex number includes multiple values because of periodicity in the complex plane, indicated by $2\pi k$.