TITLE: Does $\sqrt{i + \sqrt{i+ \sqrt{i + \sqrt{i + \cdots}}}}$ have a closed form?
QUESTION [13 upvotes]: I've been brushing up on my complex analysis recently, and I've come across a problem that's stumped me: What are the real and imaginary parts of
$$\sqrt{i+\sqrt{i+\sqrt{i+\sqrt{i+\cdots}}}} ?$$
I really don't know how to start this problem; this is my first encounter with nested roots.

REPLY [9 votes]: For the purposes of this exercise we can treat the complex square root, written in polar form like
$re^{i\phi}\mapsto \sqrt re^{i\phi/2}$, as a continuous function from the closed upper half plane $\bar{H}$ to itself, so above we specify $r\ge0,\phi\in[0,\pi]$. If $z\in\bar{H}$ then so is $i+z$, and we get a function $f:\bar{H}\to\bar{H}$ that can be abbreviated as 
$$f(z)=\sqrt{i+z}.$$
Note that by specifying the phase of the square root as above I get a well defined function.
Let 
$$D=\{z\in\bar{H}\mid \frac35\le |z|\le 2\}.$$
If $z\in D$, then $1\le|z+i|\le3$ and consequently $\frac35<\sqrt{|z+i|}<2$, so we see that restricting $f$ gives us a continuous mapping $f:D\to D$. Furthermore, $f$ is holomorphic in the interior of $D$.
To make the question precise we can study the recursively defined sequence $x_1=i$, $x_{n+1}=f(x_n)$ for all $n\ge1$. The above considerations show that $(x_n)_{n\in\Bbb{N}}$ is a sequence of elements of $D$. Because the set $D$ is compact, it has a converging subsequence (Bolzano-Weierstrass). This is the complex analysis substitute for the real analysis drill of using the theorem of bounded monotonous sequences.
Using the other answers we see that the function $f$ has a unique fixed point in $D$. Differentiating the equation 
$$
f(z)^2=i+z
$$
implies that for all $z\in D$ we have
$$
2f(z)f'(z)=1.
$$
So
$$
f'(z)=\frac1{2f(z)}
$$
has absolute value $\le\dfrac56<1$ in all of $D$. Therefore $f$ is contractible in $D$, and that unique fixed point is the limit of the entire sequence.

Of course, by specifying different branches of the square root to be used at different steps we can get other sequences with other limits.