Complex numbers can be represented in the form of a + bi, where a and b are real numbers and i is the imaginary unit (i = √(-1)). The notation z^(1/3) represents the principal cube root of z, z^(1/2) represents the principal square root of z, and ln(z) represents the natural logarithm of z. These functions are defined in the complex plane, where the real part is plotted along the x-axis and the imaginary part is plotted along the y-axis.
To show the uniqueness of these functions, we can use the argument principle, which states that the difference between the number of zeros and poles of a function inside a closed contour is equal to the number of times the function winds around the origin in the counterclockwise direction. By applying this principle to the functions z^(1/3), z^(1/2), and ln(z), we can show that they have unique values in the complex plane.
The branch cuts of these functions are the lines or curves in the complex plane where the functions are not continuous. For z^(1/3), the branch cut is along the negative real axis; for z^(1/2), the branch cut is along the negative real axis and the origin; and for ln(z), the branch cut is along the negative real axis and the origin. These branch cuts are necessary to define the functions and ensure their uniqueness.
To plot these functions, we can use the argument and modulus representations of complex numbers. The argument of a complex number z is the angle between the positive real axis and the line connecting the origin to z in the complex plane, and the modulus of z is the distance from the origin to z. By varying the argument and modulus of z, we can plot the values of z^(1/3), z^(1/2), and ln(z) in the complex plane.
Some key properties of these functions in the complex plane include their non-uniqueness (due to the presence of branch cuts), their multi-valued nature (each function has infinite possible values), and their periodicity (each function repeats its values after a certain interval in the complex plane). Additionally, these functions have different behaviors near the branch cuts and singularities, which can be studied by analyzing their derivatives and Taylor series expansions.