By using the residue theorem, prove that 1 dx 4a3 where x E R, and a >0 is a parameter.

Q: Give the condition which ensure that le*| <1 where z E C. Use the definition of the derivative of…

A: In (2.3) we have to find the condition for |ez|&lt;1. for z in C. In this part, we put z=x+iy and…

Q: Evaluate the integral O 03/20 Ⓒ100 C 87² dx using the two-point Gaussian quadrature rule. 1+16z4

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Q: Find the root ot f(x)=e"(3.25sin X-0.5 COSx) on the interval [3,4Jby using bisection method accurate…

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Q: Use Part 2 of the Fundamental Theorem of Calculus to evaluate the definite integral. a) S(1 + 3x –…

A: "Since you have asked multiple questions, we will solve the first two questions for you. If you want…

Q: 7. Evaluate using the residue theorem eir dx x2 +1

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Q: use the definition prove that s d sinZ %3D Tal

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Q: Show there is a root to the function f(z) = 9 sin z – 3x + 2 on the interval (3, 1). est tenth if…

A: Here we have, f(x)=9sinx-3x+2 ............. (1)  To prove:- The root of the function f(x)=9sinx-3x+2…

Q: sin TZ Let f (z) = Find the residue of (z – 2)? f (z)at z = 2 -

A: We have to find the residue of f(z)  at z=2

Q: By the odd extension method applied to the function f (x) = e¯2x, wherex > 0, the Fourier integral…

A: Given the function  fx=e-2x

Q: Show that u = x³ – 3xy? is a harmonic function and find a corresponding analytic function f(z) = u +…

A: To show that u=x3−3xy2 is a harmonic function and to find a corresponding analytic function fz=u+jv.…

Q: Use the Intermediate Value Theorem to show thatthere is a root of the given equation in the…

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Q: S Vx dx d.

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Q: eln (R/c) secx - sec?x)" sec(34 – x) "du?? dx to be integrable using the formula What should be the…

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Q: J32 For z = cos(x-4y) show that # дудхду 332 дугах

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Q: Suppose F(x)=∫7 to x (sin(t)/4+t) dt. Use the Fundamental of Theorem of Calculus to calculate F′(x).

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Q: By using the Residue theorem, compute the integral eiz -dz, where I is the circle |2| = 3 traversed…

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Q: Q4: (a). Prove that u = 2x(1-y) is harmonic.

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Q: dx 2 -o x+4x+5

A: The general form of a complex number is given by , where a, b are two real numbers and . Any complex…

Q: Let f(x)=sin(x)+x²-10 find Co.C,C2and c3 on the interval [1,5] using the bisection method.

A: The given function is: fx=sinx+x2-10  Also, the given interval is [1,5]. Using bisection method,…

Q: Given a function f(x,y,z) = (x+10) In (5y - 6) sin (5z). Find the domain of the function.

A: To find domain of function

Q: Use symmetry to evaluate the followingintegrals.

A: Given:∫-11 1-x dxIf f is an odd function f-x=-fx∫-aa fx dx=0If f is an even function f-x=fx∫-aafx…

Q: Find the residue of the function: f(z)=- sin(z)

A: Given fz=2sin z To find the residue of the function

Q: -Consider the function defined by the base: Find dz ) Prove that дz +y əx дz ду = sin 2z z =…

A: Topic:- application of derivatives

Q: Let u > 0 be harmonic in R². Show that u is a constant.

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Q: Find the value of the following closed integral using the residue theorem (z + 1) / (7 + 2z) dz…

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Q: Find the product of the integral 1- S x²cos(xy)dxdy

A: We have to change order of integration first.

Q: Using the Cauchy residue theorem evaluate the following improper integral cos 2 dx (22 +1)2"

A: ∫0∞cosxx2+12dx=∫0∞cosxx+i2x-i2dx Let fx=cosxx+i2x-i2. Then f has singularities at x=±i with…

Q: Given a function f(x,y,z) = (x+5) In (4y - 5) sin (10z). Find the domain of the function.

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Q: f(z) = cos-1 (포) -er s=(z) – e²

A: Concept of IVT is used while solving this problem.

Q: Use the Residue theorem to evaluate the integral fe(z-2)dz.

A: Use the expansion of e1z  =1 + 1z + 12! z2 + 13! z3 + ...  Hence ∫e1z(z-2)dz = ∫(z-2)1 + 1z + 12! z2…

Q: Given the function g(x) = :/ V4 + sin(31) dt, 2x3 use the Fundamental Theorem of Calculus Part 1 to…

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Q: 학과 학번 Determine the value(s) of d satisfying that Lacos(br+c) + d de lim 2-0c tan r dr for any real…

A: Integrate as follows.

Q: Use Residue Theorem to evaluate each of the following integrals. de (a) So 5-4 cos e dz (b) $ around…

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Q: Use the Intermediate Value Theorem to show that the following equation has at least TWOO distinct…

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Q: Find the residue of f(z) = (1 – 26) sin(1/2).

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Q: sin 7Z „Find the residue of f (z)at z = 4 (z – 4)? Let f (z) =

A: Note:

Q: Evaluate the integral cosx dx from í/3 to π/2 * 0.134 0.500 0.293 0.707

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Q: Let f(x) = e cosx, sx %3D 2. Express your answers in exact form. ElN

A: Inflection point occurs when the second derivative of the function equals to zero.

Q: Compute the residue of the following function at all singular points.  f(z)=sin(pi*z)/(z-1)3

A: Singular point of a function of the complex variable z is a point at which it is not analytic.  The…

Q: Given that f(x) = x² sinx, use Leibniz's theorem to find Ⓒd ²³ f dx²³

A: Here the given function is f(x)=x4sin(x). We have to find the third order differentiation of f(x) by…

Q: 2i z f(2)%3D at zo = 1 23 – 1 ||

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

Q: Prb $2 x sinx dx メt)

A: We have to Solve given Improper Integral using Residue Theorem as Follows :

Q: Find the minimal period T of the function: f(x) = 5 cos (rx) + sin (rx). %3D Answer: T =

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Q: Find F'(x) given that F(x) 3+sin (5t) dt. |3D In (t8) Provide your answer below: F'(x) =

A: The objective is to find F'(x).

Q: Use the Residue Theorem to evaluate the a? integral dx + x2 where a>0

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Q: Find curl F and div F for the following F=(2y-cos(x)) i -z²e3*j+(x²-7z) k

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Q: Let F(z) = | sin(5t) dt. Find the MacLaurin polynomial of degree 7 for F(z). 0.69 | sin(5z*) dz. Use…

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Q: By the odd extension method applied to the function f(x) = e-2=, where a > 0, the Fourier integral…

A: The given function is, f(x)=e-2x,x&gt;0

Q: x²7x²H4x -6 =O m Co,1그 on

A: Solution:Here x3-7x2+14x-6=0∴x3-7x2+14x-6=0Let f(x)=x3-7x2+14x-61st iteration :Here f(0)=-6&lt;0 and…