Then, write a function f(z) such that a) it is analytic in C except its isolated singular points, which are a1 and a2. b) Evaluate fler f(2)dz, where r is the average of a1 and a2.

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Q: Determine wheter f and are învers L41) f(x) = x-2, glx)=

A: solution:-Two functions f and g are inverse functions if and only ifg(f(x)) = x ,  for all x in the…

Q: Let f(z) = z2 is f(z) satisfy Cauchy-Riemann equation ?

A: Let any complex function f(z) = u + i v then it satisfies Cauchy-Riemann equation when 1) ∂u∂x=∂v∂y…

Q: Glven: XY and ZW Intersect at polnt A. Which conJecture Is always true about the glven statement?

A: as given that XY ↔ and ZW↔ intersect at point A. so, here we understand few terms which is given in…

Q: KB is ∀xy F(x, y). Prove using resolution-refutation that ∀xy F(y, x).

A: Solution: First, we convert clauses to CNF : (1) ∀xy Fx,y from KB (2) ∀x Fx,Ax (3) Fx,Ax CNF

Q: It is easy to see that u(z) = Im (+) is harmonic in the unit disk |2| <1 and lim,→1- u(reio) = 0 for…

A: Given, Uz=Im1+z1-z2,z&lt;1 Now,…

Q: sume that f(t) is continuous on [a, b| and that u = g(t) is cont ferentiable on [a, b], then f(u)…

A: Note- Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Let f be the function defined by S1+x, when 0<x<}, }<x<1 0, when x-1. f(x)= Show that fis integrable…

A: Recall the Result/Theorem A bounded function f which has only a finite number of points of…

Q: Find a potential function for F or determine that F is not conservative.〈F = z, 1, x〉

A: F is conservative when curl F = 0 Given that f=(z, 1, x)=zi+j+xk Now evaluate curl F, curl F = ∇ × F…

Q: State what it means for a scalar function y = f (x) to beintegrable on an interval [a, b].

A: Scalar integrable  function y = f (x)

Q: Integrate f(z) over unit circle C and indicate whether Cauchy's theorem may be applied (4) f(z) =…

A: With the help of Cauchy theorem of integration we solve this problem.

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…

Q: Use the Second Derivative Test to prove that if (a,b) is a critical point of f at which f,(a,b) =…

A: Since you have asked for a multiple question, we will solve the first question for you. If you want…

Q: Let H(x) integral ,for -2 ≤ x ≤2.Use geometry to evaluate H(2).

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Show that the line integral is independent of path by finding a function f such that Vf = F. 2xe-Ydx…

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Q: Show that I¿(f) = 1, and conclude that f is integrable on

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Q: Find a potential function for F or determine that F is not conservative. (If F is not conservative,…

A: Given Data The function is F=&lt;6zsec²(x), 6z, 6y+6tan(x)&gt;.   The curl of the vector field…

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A: Cauchy-Riemann Equations: Let f=u+iv be a complex valued function , then the Cauchy-Riemann…

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Q: Every function which has counatable discontinuous points on [a,b] is Riemann integrable on [a,b]. *…

A: See the solution below.

Q: Show that the line integral is independent of path by finding a function f such that Vf = F. 2xe-Ydx…

A:

Q: y = sech*x = In(_+ V1- x², sechx = In(- 1+ V1-x² Prove that .... ..

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Q: Show that (d/dz)ž does not exist anywhere, i.e., f (z) =z is non-analytic anywhere.

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Q: Verify by differentiation that a) / t dt : V1+t2 /1+t² + C; _ b) tan(x)dr = – In(cos(x)) +C, –1/2 <…

A: Given to find ∫t1+t2dtPut 1+t2=udifferentiate 2tdt=dutdt=du2therefore…

Q: Use a local linear approximation of the function f(z) = VI cos[(V2r)] at some approximate v3.98…

A: To use local linear approximation of the below function to approximate 3.98cos7.963π. fx=xcos2x3π

Q: Prove that the function f(z2) defined by x*(1+ i) – y°(1 – 1) , (z ± 0) x² + y? f(z) = [0, (z = 0)…

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Q: 4. Let f(x, y) be a function defined on a disk D not containing the origin which is given by 2.x2 f…

A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…

Q: Are the following functions Riemann Integrable to their corresponding in- tervals? Explain your…

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Q: Show that the function f(x + iy) = Vlæ|ly| satisfies the Cauchy-Riemann equations at the origin but…

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Q: The direchional derivative of the salan funchion at (l,-1,0) in the direction (i-j+2k) is xy- z- xyz…

A: Given that  Let f(x,y,z)=x^2y-y^2z-xyz To find the directional derivative at point

Q: (b) Let F = (-yx² , xy²). Let D be the annulus 1< x? + y² < 4, and let C be its boundary. Evaluate…

A: Greens theorem

Q: . Show that the function u = p(ry) + Vry v(y/x) provides the quation = 0. ду? %3D dx²

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Q: where a <c< b. Prove that he is Riemann integrable on [a, b] and he(2)dr 0. nsider the function…

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Q: Show that if f is Riemann integrable [a, b] and f(r) 0 for all r € [a, b], on f(x) dr 2 0.

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Q: Let f(x) = (cosh x)² be defined on (0, 0), find %3D d 1/2) . dy

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Q: Verify whether the function f(z) = e^x (cos y + isin y) satisfies Cauchy- Riemann equations or not.

A: We know that cauchy-Riemann equations  ux = vy  and uy = -vx Here  u = excosy and v = exsiny

Q: Suppose that f and h are integrable and that: i fx) = -1, f(x) dx = 5, h (x) dx = 4 ,Find: a. S–2…

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Q: Find the analytic function f(z) = u + iv if u = 2 cos x .cosh 2y cos 2x + cosh 2y

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Q: Show that if |sin z| is bounded along the straight line {(x, y) : ax - By = 0}, then a = 0, i.e.,…

A: We know, sin z = sin x cosh y + i cos x sinh y, Where, cosh and sinh are hyperbolic functions.

Q: Let f be the function defined by f(2) = x³ + iy³ . Determine the points z E C where f is complex…

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Q: Use the Cauchy-Riemann equations to show that f(z) = z is not analytic.

A: A function is said to be analytic when Cauchy-Riemann equations are satisfied. The Cauchy-Riemann…

Q: Consider the following theorem. Theorem If f is integrable on [a, b], then FX) dx = lim ° fx;,)ax n-…

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Q: Show that f (z) = e? is not analytic using Cauchy-Riemann equations.

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.