okay so i have searched through the entire net and i only get what is complex potential theory. But no body explain how to solve the question maybe because that's at the basic level. My question is
Find and sketch the potential. Find the complex potential too between the axes (potential 110V) and the hyperbola xy = 1 (potential 60V).
I know there is some sort of use of Laplace's Equation but i don't know how. And i even don't know how to add the values given. Kindly help needed.
Ok, so, first of all, when you refer to a book with such a vague title, you should always give also the author. Then, you could have gone back a few pages and you would have found the definition of complex potential: it is an analytic function $F$ such that $\mathrm{Re}F$ is the usual electric potential.
Now, you are given two equipotential lines: $xy=1$ and $xy=0$ (the two axes) and you need to find a function which has two given (constant) values on them and is harmonic in-between. It is quite obvious to try with something depending on $\phi(x,y)=xy$ and, if you compute $\nabla^2\phi$ you find $0$, so $\phi$ is harmonic. Now, if you set $\Phi(x,y)=Axy+B$ you get your potential having value $A$ on the hyperbola and $B$ on the axes.
To get a holomorphic function you can either solve the equations (which are explained earlier in the book) for a harmonic conjugate, or notice that the function $f(z)=z^2$ has real and imaginary part as follows $$f(x+iy)=x^2-y^2+2ixy$$ Therefore the function $$F(z)=-\frac{iA}{2}z^2 + B$$ is the complex potential you are looking for (with the convention of setting to $0$ the imaginary part of the constant term you can add to the harmonic conjugate).