A Functional Equation for the Sine

Integral Van Vleck’s and Kannappan’s functional equations on semigroups

“…when G is a not necessarily abelian group and z 0 is a fixed element in the center of G. We refer also to [11] and [12]. Perkins and Sahoo [6] replaced the group inversion by an involution τ : G −→ G and they obtained the abelian, complex-valued solutions of equation…”

The generalized Van Vleck's equation on locally compact groups

“…In the papers [8,9], Van Vleck studied the continuous solutions f : R → R, f 6 = 0, of the functional equation f (x − y + z 0 ) − f (x + y + z 0 ) = 2f (x)f (y), x,y ∈ R, (1.1) where z 0 > 0 is fixed. We shall in this paper study extensions of (1.1) and related functional equations from R to locally compact groups.…”

An extension of Van Vleck’s functional equation for the sine

“…In 1910, Van Vleck [10,11] studied the continuous solution f : R −→ R, f = 0 of the following functional equation (1.1) f (x − y + z 0 ) − f (x + y + z 0 ) = 2f (x)f (y), x, y ∈ R, where z 0 > 0 is fixed. He showed first that all solutions are periodic with period 4z 0 , and then he selected for his study any continuous solution with minimal period 4z 0 .…”