半群上的五个三角加法律
Five Trigonometric addition laws on semigroups
摘要：
本文确定了下列函数方程\[g(xσ(y)) = g(x)g(y)+f(x)f(y),\quad x,y\in S,\] \[f(xσ(y)) = f(x)g(y)+f(y)g(x),\quad x,y\in S,\] \[f(xσ(y)) = f(x)g(y)+f(y)g(x)-g(x)g(y),\quad x,y\in S,\] \[f(xσ(y))=f(x)g(y)+f(y)g(x)+αg(xσ(y)),\quad x,y\in S,\] \[f(xσ(y))=f(x)g(y)-f(y)g(x)+αg(xσ(y)),\quad x,y\in S,\]的复值解，其中$S$是半群，$α\in \mathbb{C}\backslash \lbrace 0\rbrace$是固定常数，$σ:S\rightarrow S$是对合自同构.
英文摘要：
In this paper, we determine the complex-valued solutions of the following functional equations \[g(xσ(y)) = g(x)g(y)+f(x)f(y),\quad x,y\in S,\]\[f(xσ(y)) = f(x)g(y)+f(y)g(x),\quad x,y\in S,\]\[f(xσ(y)) = f(x)g(y)+f(y)g(x)-g(x)g(y),\quad x,y\in S,\]\[f(xσ(y))=f(x)g(y)+f(y)g(x)+αg(xσ(y)),\quad x,y\in S,\]\[f(xσ(y))=f(x)g(y)-f(y)g(x)+αg(xσ(y)),\quad x,y\in S,\] where $S$ is a semigroup, $α\in \mathbb{C}\backslash \lbrace 0\rbrace$ is a fixed constant and $σ:S\rightarrow S$ an involutive automorphism.
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