How do you evaluate e^{\sqrt{i}} algebraically?

[Allday]

How do you evaluate this expression algebraically.

$$e^{\sqrt{i}}$$

 

[robphy]

Since i=exp(i*pi/2), you can determine easily sqrt(i) in polar form, then in rectangular form: a+bi. Then you can evaluate exp(a+bi).

 

[Tide]

If you take $i = e^{i \frac {pi}{2}}$ then $\sqrt i = e^{i \frac {pi}{4}}$. Just write it in trig form and put into your expression.

 

[Allday]

easy peasy. thanks

 

[existence]

Not quite. Don't forget the second root exp[i(pi/4 + pi)].

 

[Allday]

Ahh yes, those pesky multiple values. That complex plane will eventually bend to my will.