A Riemann surface is a method for visualizing multi-valued functions by treating each branch as a single-valued function on multiple sheets of the $z$-plane, which are then glued together to form a continuous transition between branches. This construction is particularly useful in the study of algebraic functions. For the function \( w = z^{1/2} \), the domain is divided into two sheets, \( S_1 \) and \( S_2 \), by cutting the \( z \)-plane along the negative \( x \)-axis. The ranges \( R_1 \) and \( R_2 \) are the right and left half-planes, respectively, with the positive and negative \( v \)-axes. These sheets are glued together along the \( v \)-axis to form the \( w \)-plane with the origin deleted. The resulting surface is a Riemann surface for \( w = z^{1/2} \). For the function \( w = z^{1/n} \), the domain is divided into \( n \) sheets, \( S_0, S_1, \ldots, S_{n-1} \), by cutting the \( z \)-plane along the negative \( x \)-axis. Each sheet is represented by a single-valued function \( f_k(z) \). The sheets are glued together in a cyclic manner, with \( s_0^- \) glued to \( s_1^+ \), \( s_1^- \) to \( s_2^+ \), and so on, up to \( s_{n-1}^- \) to \( s_0^+ \). This results in an \( n \)-sheeted Riemann surface, where the function \( w = z^{1/n} \) is single-valued and continuous for all \( z \neq 0 \).A Riemann surface is a method for visualizing multi-valued functions by treating each branch as a single-valued function on multiple sheets of the $z$-plane, which are then glued together to form a continuous transition between branches. This construction is particularly useful in the study of algebraic functions. For the function \( w = z^{1/2} \), the domain is divided into two sheets, \( S_1 \) and \( S_2 \), by cutting the \( z \)-plane along the negative \( x \)-axis. The ranges \( R_1 \) and \( R_2 \) are the right and left half-planes, respectively, with the positive and negative \( v \)-axes. These sheets are glued together along the \( v \)-axis to form the \( w \)-plane with the origin deleted. The resulting surface is a Riemann surface for \( w = z^{1/2} \). For the function \( w = z^{1/n} \), the domain is divided into \( n \) sheets, \( S_0, S_1, \ldots, S_{n-1} \), by cutting the \( z \)-plane along the negative \( x \)-axis. Each sheet is represented by a single-valued function \( f_k(z) \). The sheets are glued together in a cyclic manner, with \( s_0^- \) glued to \( s_1^+ \), \( s_1^- \) to \( s_2^+ \), and so on, up to \( s_{n-1}^- \) to \( s_0^+ \). This results in an \( n \)-sheeted Riemann surface, where the function \( w = z^{1/n} \) is single-valued and continuous for all \( z \neq 0 \).