Does $i^i$ and $i^{1\over e}$ have more than one root in $[0, 2 \pi]$

Since you probably mean the multi-valued complex exponential, you can go straight to the definition:

$$ x^y := \exp(y \log x) $$

The multi-valued complex logarithm can be computed as

$$ \log x = \ln |x| + \mathbf{i} \arg(x) $$

where $\arg(x)$ is the polar angle of $x$. This, of course, takes values in the set

$$ \arg(x) = \mathop{\text{Arg}}(x) + 2 \pi n $$

where $n$ is an integer, and $\mathop{\text{Arg}}$ is the "principal value". For your particular question, all you need to know about the idea of a principal value is that $\mathop{\text{Arg}}(x)$ is a particular choice of polar angle for $x$; here it doesn't matter which one you choose.

To demonstrate that this reproduces what you already know, consider the case of $(-8)^{1/3}$. In this case, we have:

• $\mathop{\text{Arg}}(x) = \pi$
• $\arg(x) = \pi + 2 \pi n$
• $\log(x) = \ln(8) + \mathbf{i} \pi (1 + 2n)$
• $\exp(y \log x) = \exp((1/3) \ln(8) + \mathbf{i} \pi (1 + 2n)/3) = \exp(\ln(2)) \exp(\mathbf{i} \pi (1 + 2n)/3)$
• $x^y = 2 (\cos(\frac{1+2n}{3} \pi) + \mathbf{i} \sin(\frac{1 + 2n}{3} \pi))$

and so the values for $(-8)^{1/3}$ are all of the numbers $2 \exp(\mathbf{i} \pi \frac{1 + 2n}{3})$ where $n$ is an integer. $n=1$ gives you the familiar value $-2$. $n=0$ and $n=2$ give the other two complex cube roots of $-8$. $n=4$ also gives $-2$, because $3 \pi \mathbf{i}$ and $\pi \mathbf{i}$ are the same angle. In fact, as $n$ ranges over all integers, only three distinct values are produced. This agrees with the familiar fact that nonzero complex numbers have three cube roots.

So we want a general rule for evaluating formal expressions $z^w$, where $z=r\, e^{i\phi}\ne 0$ and $w=u+iv$ are complex numbers.

Depending on circumstances one wants the full infinite set of values for $z^w$, which arises from the multivaluedness of the logarithm function in the complex domain, or one wants only the so called principal value of $z^w$. Denote by $\dot{\mathbb C}$ the punctured compex plane and by ${\mathbb C}^-$ the complex plane slit along the negative real axis. Then $\arg:\ \dot{\mathbb C}\to{\mathbb R}/(2\pi{\mathbb Z})$ is the multivalued argument function and ${\rm Arg}:\ {\mathbb C}^-\to\ ]{-\pi},\pi[\ $ the principal value of the argument. Correspondingly one has two forms of the logarithm: $$\log z:=\log|z| + i\arg(z)\ ,\qquad{\rm Log}z:=\log|z| + i{\rm Arg}(z)\ .$$

In the real domain the "general power function" $a^b$ is defined for $a>0$ by $a^b:=\exp(b\,\log a)$. Transferring this principle into the complex world one arrives at $$z^w:=\exp(w\ \log z)=\exp\bigl((u+iv)(\log r+i\phi +2k\pi i)\bigr)\qquad (r>0)\ ,$$ resp. at $$z^w:=\exp(w\ {\rm Log} z)=\exp\bigl((u+iv)(\log r + i\phi)\bigr)\qquad(r>0, \ |\phi|<\pi)\ .$$

It seems that the OP is interested in the first of the two interpretations. Therefore let's look at the example $i^i$. As $i=e^{i\pi/2}$ one has $r=1$, $\phi={\pi\over2}$. Therefore we get $$i^i=\exp\bigl( (0+i)(\log 1+i{\pi\over2}+2k\pi i)\bigr)=\exp\bigl(-{\pi\over2}-2k\pi \bigr)\ ,$$ where this is to be interpreted as the resulting set when $k$ runs through ${\mathbb Z}$. We see that this set $\{c_k\}_{k\in{\mathbb Z}}$ consists of positive real numbers. For $k\leq-1$ the $c_k$ lies outside the interval $[0,4\pi]$; but for all $k\geq0$ one has $c_k\in[0,4\pi]$, so that, yes: there are infinitely many values of $i^i$ in the given interval.

Let $e^z=N≠0$ ,So, $N=e^z=e^z\cdot e^{2n\pi i}$ as $e^{2n\pi i}=1$ for any integer $n$.

$N=e^{z+2n\pi i}\implies Log N=z+2n\pi i$

If $n=0$ ,we get the principal value $z=\log N$, so $LogN=\log N+2n\pi i$

Let $A=(a+ib)^{x+iy}$

$Log A=(x+iy)Log(a+ib)$

Let $a=r\cos \theta$ and $b=r\sin \theta\implies a+ib=r(\cos \theta+i\sin \theta)=re^{i\theta}$

So, $r=\sqrt{a^2+b^2}$ and $\theta=\tan ^{-1}\frac{b}{a}$

then $Log(a+ib)=\frac{1}{2}\log (a^2+b^2)+i(2k\pi+\tan ^{-1}\frac{b}{a})$ where $k$ is any integer.

So, $Log A=(x+iy)Log(a+ib)=(x+iy)(\frac{1}{2}\log (a^2+b^2)+i(2k\pi+\tan ^{-1}\frac{b}{a}))$

So, $Log A=\frac{1}{2}x\log (a^2+b^2)-y(2k\pi+\tan ^{-1}\frac{b}{a})+i(\frac{1}{2}y\log (a^2+b^2)+x((2k\pi+\tan ^{-1}\frac{b}{a}))$

So, $$A=(a+ib)^{x+iy}=e^{\frac{1}{2}x\log (a^2+b^2)-y(2k\pi+\tan ^{-1}\frac{b}{a})}\cdot e^{i({\frac{1}{2}y\log (a^2+b^2)+x(2k\pi+\tan ^{-1}\frac{b}{a}})}$$

$$=e^{\frac{1}{2}x\log (a^2+b^2)-y(2k\pi+\tan ^{-1}\frac{b}{a})} cis(({\frac{1}{2}y\log (a^2+b^2)+x(2k\pi+\tan ^{-1}\frac{b}{a}})) $$ where $cis(P)=\cos P+i\sin P$

(1)For $(1+i)^i$ $a=b=y=1$ and $x=0$

So, $r^2=a^2+b^2=2$ and $\sin \theta=\cos \theta=\frac{1}{\sqrt2}\implies \theta=\frac{\pi}{4}$

$(1+i)^i=e^{-(2k\pi+\frac{\pi}{4})}(\cos (\frac{1}{2}\log 2)+i\sin (\frac{1}{2}\log 2))$ where $k$ is any integer.

(2)For $(1+i)^e, a=b=1$ and $x=e,y=0$

So, $r^2=a^2+b^2=2$ and $\sin \theta=\cos \theta=\frac{1}{\sqrt2}\implies \theta=\frac{\pi}{4}$

$(1+i)^e=e^{\frac{1}{2}e\log 2}\cdot cis(e(2k\pi+\frac{\pi}{4}))$ where $k$ is any integer.

$(1+i)^e=2^{(\frac{e}{2})}\cdot cis(e(2k\pi+\frac{\pi}{4}))$

(3) For $(1+i)^{\frac{i}{e}}, a=b=1$ and $x=0,y=\frac{1}{e}$

$(1+i)^{\frac{i}{e}}=e^{-\frac{1}{e}(2k\pi+\frac{\pi}{4})}\cdot cis(\frac{1}{2e}\log 2)$ where $k$ is any integer.

(4) For $i^i,a=0,b=1,x=0,y=1\implies r^2=1$ and $\sin \theta=1$ and $\cos \theta=0\implies \theta=\frac{\pi}{2}$

So, the general value of $i^i=e^{-(2k\pi+\frac{\pi}{2})}$ where $k$ is any integer.

Now by the given condition, $0 ≤e^{-(2k\pi+\frac{\pi}{2})} ≤4\pi$

$0 ≤e^{-(2k\pi+\frac{\pi}{2})}$ is true for all real finite $k$.

So, we need check $ -(2k\pi+\frac{\pi}{2})≤\ln _e{4\pi}$ .

or $2k\pi+\frac{\pi}{2}≥-\ln _e{4\pi}$

Without using calculator, we can observe that this inequality is satisfied by all integer $k≥0$

(5) For $i^{\frac{1}{e}}, a=0,b=1,x=\frac{1}{e},y=0$ $\implies r^2=1$ and $\sin \theta=1$ and $\cos \theta=0\implies \theta=\frac{\pi}{2}$

$i^{\frac{1}{e}}=e^{\frac{1}{2e}\log 2}\cdot cis(\frac{1}{e}(2k\pi+\frac{\pi}{2}))$

can be further simplified to $2^{\frac{1}{2e}}\cdot cis(\frac{1}{e}(2k\pi+\frac{\pi}{2}))$ where $k$ is any integer.

It does have an imaginary part, so can not have any pure real value in any specified range.