{\displaystyle \log(\cos \theta +\operatorname {sen} \theta {\sqrt {-1}})^{k}=(k\theta \pm 2mk\pi \pm n\pi ){\sqrt {-1}}}

Given the expression 

\[ \log(\cos \theta + \sin \theta \cdot i)^{k}=(k\theta \pm 2mk\pi \pm n\pi )\cdot i \]

where \( \operatorname{sen} \) is likely a typographical error and should be \( \sin \).

We can analyze and simplify this expression step-by-step:

1. **Argument of the complex number:** 
   \[ \cos \theta + \sin \theta \cdot i \]
   This complex number can be represented in exponential form using Euler's formula:
   \[ \cos \theta + \sin \theta \cdot i = e^{i\theta} \]

2. **Raise to the power \( k \):**
   \[ (e^{i\theta})^k = e^{i k \theta} \]

3. **Logarithm of the complex number:**
   \[ \log(e^{i k \theta}) = i k \theta + 2m \pi i \]
   Here, \( 2m\pi i \) is added to account for the multi-valued nature of the complex logarithm, where \( m \) is any integer.

Therefore, the expression becomes:
\[ \log(\cos \theta + \sin \theta \cdot i)^k = \log(e^{i k \theta}) = i k \theta + 2m \pi i \]

4. **Considering all values (plus or minus \( n\pi \)):**
   Since \( e^{i\theta} \) can represent a rotation, we can also add \( \pm n \pi i \) to account for possible phase shifts.

Thus, the expression simplifies to:
\[ \log(\cos \theta + \sin \theta \cdot i)^k = (k\theta \pm 2mk\pi \pm n\pi )i \]

This concludes the derivation.

Would you like more details or have any questions on any step?

Here are 5 questions you might want to ask:
1. Can you explain the Euler's formula used in the derivation?
2. How do complex logarithms work and why do we add \(2m\pi i\)?
3. What does the expression represent in terms of complex functions?
4. Can you provide an example with specific values for \( \theta \) and \( k \)?
5. How does the multi-valued nature of the logarithm affect the solution?

**Tip:** When dealing with complex numbers, converting to exponential form often simplifies multiplication and logarithmic operations.

Explore the derivation of the complex logarithm for the expression (cos θ + sin θ i)^k. This problem covers Euler's formula, the multi-valued nature of complex logarithms, and detailed steps for simplification. Ideal for undergraduate students studying complex numbers and logarithms.

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