If is entire such that in then show t

Question

If $f = u + iv$ is entire such that $u_x + v_y = 0$ in $\mathbb{C}$ then show that f has the form $f(z) = az + b$ where $a, \mathbf{b}$ are constants with $\operatorname{Re} a = 0$ .

09 Jan 2026

Answer :

Word Count : 149

Let (f=u+iv) be entire. Then (u) and (v) have continuous first partial derivatives everywhere and satisfy the Cauchy–Riemann equations [ u_x=v_y,\qquad u_y=-v_x . ] The given condition (u_x+v_y=0) together with (u_x=v_y) yields [ u_x+u_x=0 \quad\Rightarrow\quad u_x=0, ] _________ _______ ______ ____ ______ __________ __________ _________ _______ _____.
_______ ____ __________ ________ ______ ________ _____ _____ ________.
_________ _________ ________ ____ ________ _______ __________ __________ _______ ________ ________.
___ ______ _____ _____ ________ _______ ___ ______ __________.
_________ ________ _________ ___ _________ __________ ______ _____ __________ ____ ________ _____.
________ _________ _____ ________ __________ _____ _____ __________ ______ _____ ____.
__________ _______ ________ ___ _______ ______ ___ ______.
___ ________ ____ _________ __________ _____ ______ _____.
___ ___ _________ ____ ____ ___ _________ _______ _____.
_____ __________ __________ ____ _____ ___ ____ _________ __________.
______ __________ _______ __________ _____ ________ _____ _______.
_________ __________ ____ ___ ____ ___.
_________ ________.
Get Full Answer on WhatsApp