Example 7. Show that the function f(z) = u + iv, where f(2)= x³(1+i)-y³(1 - i) x² + y² is continuous and that Cauchy-Riemann equations are satisfited at the origin, yet f'(0) does not exist. ,2 #0 and f(0) = 0

Q: 2. Let u,v be functions of x. The first order derivative of uv can be derived by the following…

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Q: 8. Sketch a graph of f(x) that satisfies all of the following conditions: i) f(x) is continuous and…

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Q: d g(s) be two differentiable functions in R and f  (2) = 8, g(2) =0, f(4) = 10 , and g(4) = 8 Then…

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Q: f(x + h) – f(x) f'(x) = lim h-0 %3D h bsnd Diffeentinble A) 7xh2 B) 9x2h C) 5x2 D) 9x2 5) Given (x")…

A: Given that: 4. f(x+h)-f(x)=9x2h+7xh2+5h35. (xα)'=αxα-1

Q: t f(x+y)=f(x)+f(y) + 2xy-1 for all real x and y and f(x) be a differentiable function. [f f^ prime…

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Q: For what value of the constant cis the function f continuous on(-0,0) x2+ 2x +6 if xsZ cosin (x-2)…

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Q: Prove tht the fallowing ralt U(1-k)-2U(t) + U(t + k) U" (t) =- with an enor of ortler k?, which is…

A: The objective is to prove the central difference approximation to the second derivative of Ut…

Q: (C) Using Cauchy-Riemann theorem, show that f' and f" exist everywhere, then find f' and f", where…

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Q: 2. Use the theorem in Sec. 23 to show that f'(z) and its derivative f"(z) exist everywhere, and find…

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Q: 3. Show that the function f(2)=√Im(2²) satisfies the Cauchy-Riemann equations at z = 0, but is not…

A: Given: fz=Imz2. To Do: Show that the given function satisfies the Cauchy-Riemann equations at z=0,…

Q: Show that f(z) = x ² + y ² where z = x + iy is not analytic anywhere using Cauchy Riemann equations.

A: Cauchy Riemann Equations: Given,  f(x+iy)=u(x,y)+iv(x,y)  and it should satisfy  ∂u∂x=∂v∂y and…

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Q: Satisfies the Cauchy-Riemann equations at point z=0. Is the function analytic at z=o? Justify your…

A: By the sufficient condition of the analytic function or differentiability of the complex function,…

Q: Prove that if f'(z)=0 then f(z) is a constant.

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Q: f(1) = 4 s' (1) = -2 g (3) = 7 gʻ (3) = 1 The point (1, 3) lies on the curve in the xy-plane given…

A: f(1) = 4 f'(1) = -2 g(3) = 7 g'(3) = 1 f(x)g(y) = 24 + x + y

Q: y The function f(x, y)= (2x* + y* )exp + xy' is homogenous in x and y of degree 2x

A: Follow the procedure given below.

Q: f(r, y, z) = sec='(9x + 3yz) f. = fy = f; =

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Q: 4 a. Consider the i.v.p x' = t^(2) + cos(x), x(0) = 0. Verify that the hypothesis of Cauchy Picard…

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Q: Let g(x) = 2 f(u) + 3 where . u= u = with respect to x? ○ 2. f' (u) +3 O (2. f' (u) + 3) (2x + 2)…

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Q: Suppose f: R → R has three derivatives and also satisfies the following three conditions: (1) There…

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Q: If f (x) = 2 x³ + 8 x² + 3, at what points are f '(x) zero or does not exist ? 8 0, 3 A 8 B 0, 3 1,…

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Q: Show that the function f(x) = x’ + 1 is invertible from R to R.

A: First prove that the  function is one -one  let f(x)=fyx7+1=y7+1x7=y7x=y therefore, the function is…

Q: Suppose f1 and f2 are two analytic functions so that Re(f1)=u and Re(f2)=u.  Then h(z)=f1(z)-f2(z)…

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Q: By direct differentiation of x(t) = eAtxo + eA(t-o) Bu(o)do show that ¿(t) = Ax(t) + Bu(t), x(0) =…

A: The differentiation of a definite integral can be obtained by using the Chain Rule following…