= Constant-modulus holomorphic function
A <holomorphic function> with constant modulus on a connected <domain> is constant. If $h=u+iv$ and $u^2+v^2=c^2>0$, differentiation and the <Cauchy-Riemann equations> give $uu_x+vv_x=0$ and $-uv_x+vu_x=0$. Their coefficient <determinant> is $-c^2$, so $u_x=v_x=0$, and all first <derivatives> vanish. When $c=0$, the conclusion is immediate. Consequently two nonvanishing <holomorphic functions> with equal moduli differ by a single constant phase on a connected <domain>.
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