A Riemann surface is a method to visualize multi-valued functions by treating all branches as a single-valued function on a domain composed of multiple sheets of the z-plane. These sheets are connected so that moving between them continuously transitions between different branches of the function. This structure is called a Riemann surface for the function. For the function $ w = z^{1/2} $, there are two branches, represented by $ f_1(z) $ and $ f_2(z) $. Each has a domain obtained by cutting the z-plane along the negative x-axis. The ranges of these functions are the right and left half-planes along with the positive and negative v-axes, respectively. These ranges are glued together to form the w-plane with the origin deleted. By placing the two cut z-plane sheets directly on each other, the upper sheet maps to the right half-plane and the lower sheet to the left half-plane. The edges of the sheets are glued together, forming a Riemann surface for $ w = z^{1/2} $. On this surface, the function is single-valued and continuous for all $ z \neq 0 $. For the function $ w = z^{1/n} $, there are n branches, each represented by a single-valued function. The domains of these functions are obtained by cutting the z-plane along the negative x-axis. These n sheets are then connected in a cyclic manner, with the edge of one sheet glued to the edge of the next. This results in an n-sheeted Riemann surface. All sheets meet at the branch point z=0. On this surface, the function $ w = z^{1/n} $ is single-valued and continuous for all $ z \neq 0 $.A Riemann surface is a method to visualize multi-valued functions by treating all branches as a single-valued function on a domain composed of multiple sheets of the z-plane. These sheets are connected so that moving between them continuously transitions between different branches of the function. This structure is called a Riemann surface for the function. For the function $ w = z^{1/2} $, there are two branches, represented by $ f_1(z) $ and $ f_2(z) $. Each has a domain obtained by cutting the z-plane along the negative x-axis. The ranges of these functions are the right and left half-planes along with the positive and negative v-axes, respectively. These ranges are glued together to form the w-plane with the origin deleted. By placing the two cut z-plane sheets directly on each other, the upper sheet maps to the right half-plane and the lower sheet to the left half-plane. The edges of the sheets are glued together, forming a Riemann surface for $ w = z^{1/2} $. On this surface, the function is single-valued and continuous for all $ z \neq 0 $. For the function $ w = z^{1/n} $, there are n branches, each represented by a single-valued function. The domains of these functions are obtained by cutting the z-plane along the negative x-axis. These n sheets are then connected in a cyclic manner, with the edge of one sheet glued to the edge of the next. This results in an n-sheeted Riemann surface. All sheets meet at the branch point z=0. On this surface, the function $ w = z^{1/n} $ is single-valued and continuous for all $ z \neq 0 $.