Confusion about exponential function. What are the differences between e^x and e^ix both graphically and theoretically?And my second question is what is the precise intuition behind rotation of 2D number and exponential function?
 A: Hint: $e^{ix}=\cos x + i\sin x$. This is called the Euler's formula and sometimes written as $\text{cis } x$.
A: Well, if you're allowing complex valued $x$ there isn't a whole lot of theoretical difference. Just have $f(x)\to f(ix).$ One is just a "rescaled" version of the other (I use quotes cause in the complex plane "rescaling" x by $i$ is the same as rotating by 90 degrees. See more on this below.).
However if you're restricting to a real-valued $x$ then there's a world of difference. $e^x$ is a real valued function that you can graph simply. Famously, it is a positive, concave up curve that increases exponentially as $x\to \infty$ and asymptotes to zero as $x\to-\infty.$
On the other hand $e^{ix}$ can be written $\cos(x)+i\sin(x),$ so is a complex-valued function. If you watch the value on the complex plane as you increase $x$ you will see it wind and wind around the unit circle in a counterclockwise fashion. This is related to the parametrization of the unit circle $(x(t),y(t)) = (\cos(t),\sin(t)).$
If you multiply a complex number $z$ by $e^{ix},$ you can understand what's happening by writing the complex number in polar coordinates: $z=re^{i\theta} = r\cos(\theta) + ir\sin(\theta).$ Then we have $$ ze^{ix} = re^{i\theta}e^{ix} = re^{i(\theta+x)}$$ (To see this you can either trust the rules of exponentiation work in this complex territory or write everything out as sines and cosines and use trig identities.)
Thus the effect of multiplying by $e^{ix}$ for $x$ real is to add $x$ to the polar angle coordinate of the number, or, in simpler terms to rotate it counterclockwise by angle $x$. 
A: Multiplication by $e^{i\theta}$ (where $\theta$ is real) corresponds to a rotation of $\theta$ about the origin. This is best checked by using the polar form $z=r\cdot e^{i\theta}$. Perhaps a more basic approach is using Euler's formula
$$e^{i\theta}=\cos(\theta)+i\sin(\theta)$$
and trigonometry (angle sum formulas) to justify why multiplication in polar form works the way it does. If you want to go even deeper, however, you will probably need to use power series to justify Euler's formula itself.
Now, since
$$i=\exp\left({\displaystyle i\frac{\pi}{2}}\right),$$
we have that $ix$ is obtained from $x$ by a $90^\circ$ degree rotation. The maps $e^{ix}$ and $e^x$ are hence related by this rotation.
