The equation you provided is:

$z = i^i + z \cdot i$

Let's break it down and solve it step by step.

1. First term: $i^i$

   • The imaginary unit $i$ is defined as $\sqrt{-1}$. When raised to itself, $i^i$ is a complex number.
   • Using the Euler's formula, $i = e^{i\frac{\pi}{2}}$, we can compute $i^i$.

   $i^i = e^{-\frac{\pi}{2}} \approx 0.20788$

   So, $i^i$ is a real number.

2. Second term: $z \cdot i$

   • This represents the product of the complex variable $z$ and the imaginary unit $i$. If $z = a + bi$, multiplying by $i$ gives:

   $z \cdot i = (a + bi) \cdot i = ai + b(-1) = -b + ai$

Thus, we now have:

$z = 0.20788 + (-b + ai)$

This gives the complex number form of $z$. To find specific values for $a$ and $b$, you'd need additional conditions or context for the value of $z$.

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Do you need further details or have any questions?

Here are 5 related questions to explore:

1. What is Euler's formula and how is it applied to compute ( i^i \