Show that an analytic function can not have a constant modulus without reducing to a constant.

Complex Analysis Theorem from Analytic function Statement/Theorem /Prove that :- Show that an analytic function with constant modulus is constant. OR Show that an analytic function cannot have a constant modulus without reducing to a constant. Solution/ Proof:- Let a function f(z)= u+iv According to Cauchy-Riemann equations (∂u/∂x)=(∂v/∂y) and (∂u/∂y)=-(∂v/∂x) ........(1) The modulus of the function f(z)=u+iv ⇒|f(z)|=constant = c (say) ⇒√(u²+v²)=c ⇒u²+v²=c² ........(2) Now Partially differentiating Equation (2) w.r.t. x and y 2u(∂u/∂x)+2v(∂u/∂x)=0 ⇒u(∂u/∂x)+v(∂u/∂x)=0 ∵(∂v/∂x)=-(∂u/∂y) ⇒u(∂u/∂x)-v(∂u/∂y)=0 ........(3) And 2u(∂u/∂y)+2v(∂u/∂y)=0 ⇒u(∂u/∂y)+v(∂u/∂y)=0 ∵(∂v/∂y)=(∂u/∂x) ⇒u(∂u/∂y)+v(∂u/∂x)=0 ........(4) Multiplying v and u in question (3) and (4) respectively and subtracting Equation (4) from equation (3) we get vu(∂u/∂x)-v²(∂u/∂x)-vu(∂u/∂y)-v²(∂u/∂x)=0 ⇒-2v²(∂u/∂x)=0 ⇒(∂u/∂x)=0 Putting in equation (3) we get ⇒(∂u/∂y)=0 And by( Cauchy-Riemann) equations (1) ⇒(∂u/∂x)=(∂v/∂y)=0 and ⇒-(∂u/∂y)=(∂v/∂x)=0 Here (∂u/∂x),(∂u/∂y),(∂v/∂x)and(∂v/∂y)are equal to zero. This shows that u and v are constant and function is f(z)=u+iv = constant. Hence Proved...!!! . . . . . . Kindly join us for CSIR-UGC National Eligibility Test (NET) for Junior Research Fellowship and Lecturer-ship COMMON SYLLABUS FOR PART 'B' AND 'C' MATHEMATICAL SCIENCE . . . . . All BSc, B.Tech, MSc, MA Mathematics students join with us A great platform to get knowledge in Mathematics and science... Stay tuned....for detailed videos on whole SYLLABUS with topic wise... . . . . . . . #complexanalysis #csirnet #analyticfunction . . Videos links 🔥🔥🔥 1)    • Coming Soon With A New Concept of Learning...   2)    • Continuity is necessary but not sufficient...   3)    • Cauchy-Riemann Equation : (C-R Equation ) ...   4)    • The Sufficient Condition for an Analytic F...   5)    • The Polar Form of Cauchy-Riemann equation ...   6)    • The Real Valued function of a complex vari...   7)    • If n is real, r^n(cosnθ+isinnθ) is Analyti...   8)    • Φ(x,y) & ψ(x,y) are satisfied Laplace eq'n...   9)    • Laplace’s Equation: f(z)= u+iv Analytic fu...   10)    • Analytic Function: Show that f(z) = z̄ is ...   11)    • Real & Imaginary part of an Analytic funct...   12)   • Show f(z) = √|xy| is not analytic at origi...   13)   • Show f(z)={x³(1+i)-y³(1-i)}/(x² + y²) ,f(0...   Play List ( Complex Analysis) 👉    • Complex Analysis