I noticed that you had the step of the attempt that: $ππ¦^{ππ¦}$=$π^{ππ¦πππ(ππ¦)}$=$π^{ππ¦}βπ^{πππ(ππ¦)}$ when this step should be, based on my knowledge of real number exponential rules:
Using exp(x)=$e^x$:
$π^{ππ¦πππ(ππ¦)}$=$exp(iy)^{log(iy)}$= $(exp({log(iy)))^{iy}}$.
Exponents multiplied are an exponent to the power of another exponent whereas multiplications of the same bases would get you added exponents: $π^{ππ¦}βπ^{πππ(ππ¦)}$= $π^{ππ¦+πππ(ππ¦)}$.
Anyways, as exp(iΟ)=-1,$\sqrt {exp(iΟ)}=\sqrt {-1}=i=$exp($\frac{Οi}{2})$, this means that one value of ln(i) is $\frac{Οi}{2}$. I will also use log(x)=ln(x) for convenience:
exp(iy*log(iy))=exp(iy(ln(i)+ln(y))=$exp(iy(\frac{Οi}{2}+ln(y))$= $exp(\frac{-Οy}{2}+iy*ln(y))$= $exp(iy*ln(y)-\frac{Οy}{2})$
Splitting the exponent into two different bases reveals that:
$exp(iy*ln(y)-\frac{Οy}{2})$=$ exp(-\frac{Οy}{2})*exp(i(y*ln(y)))$=$C*e^{iΞΈβ}$ where C=$e^{-\frac{Οy}{2}}$ and ΞΈ-ΞΈβ=2Οn,n $\in \Bbb Z$. Here ΞΈβ= y*ln(y)
Using Eulerβs Identity and/or De Moivreβs Theorem shows the final answer is:
$f=f(y)=(iy)^{iy}=C(cosΞΈβ+i*sinΞΈβ)=e^{-\frac {Οy}{2}} cos(y*ln(y)+2Οn)+i e^{-\frac {Οy}{2}} sin(y*ln(y)+2Οn)$
= Re(f)+i*Im(f), nββ€.
This can also be written as: $e^{-\frac {Οy}{2}} [cos(y*ln(y)+2Οn)+i sin(y*ln(y)+2Οn)]$, nββ€.
Letβs try y=1:
f(1)=$i^i$= $e^{-\frac {Ο}{2}} [cos(1*ln(1)+2Οn)+i sin(1*ln(1)+2Οn)]$ and n=0:
f(1)= $e^{-\frac {Ο}{2}}[1+0i]$=$e^{-\frac {Ο}{2}}$
Letβs try y= $-\frac {i}{2}$:
f(-$\frac {i}{2}$)= $e^{-\frac {Ο\frac {-i}{2}}{2}} [cos((-\frac {i}{2})*ln(-\frac {i}{2})+2Οn)+i sin((-\frac {i}{2})*ln(-\frac {i}{2})+2Οn)]$
Using a similar technique to find $ln(-\frac {i}{2})$ and letting n=0, we get:
f(-$\frac {i}{2}$)= $(i\frac {-i}{2})^{(i \frac {-i}{2})}$=$(\frac 12)^{\frac 12}$=$\frac{1}{\sqrt2}$=$(\frac{1+i}{\sqrt2})[cos(\frac 14(2i*ln(2))+isin(\frac 14(2i*ln(2))]$
This answer simplifies down to:
$cosh(\frac{ln(2)}{2})-sinh(\frac{ln(2)}{2})$ which uses the hyperbolic trigonometric functions.
This finally simplifies down to $e^{-\frac{ln(2)}{2}}=\frac{1}{\sqrt 2}$ if the hyperbolic functions are written in terms of their exponential definition.
Here is a link about them:
https://en.m.wikipedia.org/wiki/Hyperbolic_functions
Correct me if I am wrong!
Best Answer
$$\arg(z-3-2i)=\frac{\pi}{6}$$ is a line which originate from $(3,2)$ and making an angle of $30^\circ$ with positive $x$ axis.
And $$\arg(z-3-4i)=\frac{2\pi}{3}$$ is a line which originate from $(3,4)$ and making an angle of $120^\circ$ with positive $x$ axis.
Now drawing These line in $x-y$ Coordinate axis.
You will get no point of Intersection.
So no $z$ which satisfy above these two equations.