sinImz

sinImz is a mathematical designation used to denote the real-valued function on the complex plane defined by f(z) = sin(Im z). If z = x + i y, then Im z = y and f(z) = sin(y). In other words, the function depends only on the imaginary part of z and is independent of the real part x.

Properties of sinImz include that it is real-valued for all z, continuous and infinitely differentiable with respect to y, and periodic in y with period 2π. Its range is [-1, 1], and along any horizontal line where Im z is fixed, the value is constant as z varies in the real direction.

Analytic status: sinImz is not holomorphic on any nonempty open subset of the complex plane. Writing z = x + i y with f(z) = u(x,y) + i v(x,y) gives u(x,y) = sin(y) and v(x,y) = 0. The Cauchy-Riemann equations require u_x = v_y and u_y = -v_x; here u_x = 0 and u_y = cos(y), while v_x = v_y = 0. The second equation would demand cos(y) = 0, which holds only at isolated lines, not on any open set. Therefore sinImz is not analytic anywhere in the plane.

Visualization and variants: Graphically, the surface f(x,y) = sin(y) forms horizontal sine stripes, extending along the real axis. A related idea is sin(Re z) = sin(x), which depends on the real part instead. The term sinImz is informal and mainly used to illustrate non-analytic behavior in contrast with holomorphic functions. See also complex analysis, Cauchy-Riemann equations, and holomorphic functions.