Show f(z)={x³(1+i)-y³(1-i)}/(x² + y²) ,f(0) = 0 continuous & C-R eq are satisfy yet f'(0) not exist.

Complex Analysis Theorem from Analytic function Statement/Theorem /Prove that :- Show that the function f(z) = u+iv, where f(z) = {x³(1+i)-y³(1-i)}/(x² + y²) , z ≠ 0 and f(0) = 0 is continuous and that Cauchy-Riemann equations are satisfited at the origin, yet f'(0) does not exist. Solution/ Proof:- We have f(z) = {x³(1+i)-y³(1-i)}/(x² + y²) ⇒u+iv = {(x³-y³)+i(x³+y³)}/(x² + y²) ⇒u= (x³-y³)/(x² + y²) and v=(x³+y³)/(x² + y²) By definition of Continuity lim z→z₀ f(z) = lim z→z₀ {x³(1+i)-y³(1-i)}/(x² + y²) lim z→z₀ f(z) = lim ₓ→ₓ₀ , ᵧ→ᵧ₀ {x³(1+i)-y³(1-i)}/(x² + y²) lim z→z₀ f(z) = {x₀³(1+i)-y₀³(1-i)}/(x₀² + y₀²) lim z→z₀ f(z) = f(z₀) Hence f(z) is continuous. Now at origin (0,0) (∂u/∂x)= lim ₓ→₀ {u(x,0)-u(0,0)}/x = lim ₓ→₀ (x³/x²)/x = lim ₓ→₀ x³/x³ = 1 (∂u/∂y)= lim ᵧ→₀ {u(0,y)-u(0,0)}/y = lim ᵧ→₀ (-y³/y²)/y = lim ᵧ→₀ (-y³/y³) = -1 (∂v/∂x)= lim ₓ→₀ {v(x,0)-v(0,0)}/x = lim ₓ→₀ (x³/x²)/x = lim ₓ→₀ x³/x³ = 1 (∂v/∂y)= lim ᵧ→₀ {v(0,y)-v(0,0)}/y = lim ᵧ→₀ (y³/y²)/y = lim ᵧ→₀ y³/y³ = 1 Here (∂v/∂x)=(∂v/∂y) and (∂v/∂x)=-(∂u/∂y) ∴ u and v satisfy Cauchy-Riemann equations at origin. Now for existence of differentiability f'(0)=lim z→o {f(z)-f(0)}/(z-0) f'(0)=lim z→o {(x³-y³)+i(x³+y³)}/{(x² + y²)(x+iy)} Along real axis z=x , y=0 ⇒z→0 ∴ x→0 f'(0)=lim ₓ→₀ {(x³-0)+i(x³+0)}/x(x² +0) =lim ₓ→₀ (x³+ix³)/x³ =(1+i) Along imaginary axis z=iy , x=0 ⇒z→0 ∴ y→0 f'(0)=lim ᵧ→₀ {(0-y³)+i(0+y³)}/(iy)(0 + y²) =lim ₓ→₀ (-y³+iy³)/iy³ =(-1+i)/i =(1+i) Along a line curve y=x f'(0)= lim z→o {(x³-x³)+i(x³+x³)}/{(x² + x²)(x+ix)} f'(0)= lim z→o (0+i2x³)/{2x³(1+i)} f'(0)= lim z→o (i2x³)/{2x³(1+i)} f'(0)= i/(1+i) f'(0)= (i+1)/2 This shows that f'(0) has different values Along difference curve ∴ f'(z) is not unique , Hence f'(0) does not exist. 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...   Play List ( Complex Analysis) 👉    • Complex Analysis