fminsearch is a very general optimization function which you could use to minimize the error, i.e. to minimize $|f(b) - 0|$. If a solution exists, then that norm expression should be 0. It is a very useful function if you have a small and simple problem you need a fast solution to.
However, as your function is differentiable with respect to the variables, you may want to use a numerical scheme like fzero that Chinny84 proposed above.
Both of these functions you need to know how to pass on functions to other functions:
f = @(b) 2*I*sinh((a+b)/(2*V)) - (a-b)/(2*R);
Then you pass it on as :
b_solution = fzero(f,$b_0$);