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The question asks to find the general and principal value of \( (1 + i)^{-i} \), where \( i \) is the imaginary unit.

### Step-by-Step Solution:

1. **Express \( 1 + i \) in polar form**:
   The complex number \( 1 + i \) can be written as \( r (\cos \theta + i \sin \theta) \) in polar form.
   
   First, find the modulus \( r \) of \( 1 + i \):
   \[
   r = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2}
   \]

   Now, find the argument \( \theta \):
   \[
   \theta = \arg(1 + i) = \tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4}
   \]

   So, \( 1 + i \) in polar form is:
   \[
   1 + i = \sqrt{2} \left( \cos\frac{\pi}{4} + i\sin\frac{\pi}{4} \right)
   \]

2. **Find \( (1 + i)^{-i} \)**:
   Now we need to compute \( (\sqrt{2} \text{cis} \frac{\pi}{4})^{-i} \), where \( \text{cis} \theta = \cos \theta + i \sin \theta \).

   According to the properties of complex exponentiation:
   \[
   z^{-i} = r^{-i} \text{cis}(-i \theta)
   \]

   Therefore:
   \[
   (\sqrt{2} \text{cis} \frac{\pi}{4})^{-i} = (\sqrt{2})^{-i} \text{cis} \left( -i \cdot \frac{\pi}{4} \right)
   \]

3. **Simplify the modulus**:
   The modulus part \( (\sqrt{2})^{-i} \) can be simplified using the formula \( a^b = e^{b \ln a} \):
   \[
   (\sqrt{2})^{-i} = e^{-i \ln (\sqrt{2})} = e^{-i \cdot \frac{1}{2} \ln 2} = e^{\frac{\ln 2}{2}}
   \]

4. **Simplify the argument**:
   The argument part becomes:
   \[
   \text{cis} \left( -i \cdot \frac{\pi}{4} \right) = \text{cis} \left( \frac{\pi}{4} i \right)
   \]
   This gives the general value of the expression.

### Final Answer:
\[
(1 + i)^{-i} = e^{\frac{\ln 2}{2}} \cdot \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)
\]
This is the expression for the general value of \( (1 + i)^{-i} \). The principal value is the same, as it involves the same argument of \( \frac{\pi}{4} \).

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Would you like more details on how to simplify further or how to find similar complex values?

Here are 5 related questions for practice:
1. How do you express a general complex number in polar form?
2. What is the significance of the principal and general values in complex numbers?
3. How do you find the modulus and argument of a complex number?
4. How can you apply Euler's formula to compute powers of complex numbers?
5. What is the difference between the principal argument and the general argument of a complex number?

**Tip**: Always check the quadrant of the complex number when determining the argument to avoid errors.

Find the general and principal value of (1 + i)^{-i}.

Explore how to find the general and principal value of (1 + i)^{-i} using polar form, complex exponentiation, and Euler's formula. This problem covers key concepts in complex numbers, exponential forms, and related mathematical theorems. Ideal for advanced high school or college-level students.

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