21. Given that z is a complex number satisfying |2z 1 |z - 2|, prove that |z| (a) letting z = x + iy, (b) squaring the equation and then using the result z2 1 by: = 2z.

Q: 4. Let z = -4 + 3i and zw = -7 – 2i. By showing clear steps, find w in the form of a+bi.

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Q: Show that i + (±)³ + x 0 has no periodic solution.

A: Given: To show that the given   has no periodic solution.

Q: 1) Prove that there are no positive integer solutions to the equation x2-y?=1

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Q: 2.3 Find the general solution of p fanx+q tany =lanz

A: Given:   P tanx+q tan y=tanz→           1   is of the form pP+qQ=z:

Q: 17. Given a = 4 – 4i and az = 4 + 3i, express %3D :11 in the form a + bi. a, az CLONE Dec 19

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Q: 2.3 Evaluate the integral sinh 2 dz, 7 3i (z – 2i)4 2i 2+ 4 where C is the circle |z – 2i| = 4…

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Q: 2. Solve the following Cauchy problem: Uu = Uzz -0 0, u(x,0) = x – a², u(x,0) = 4x, -00 <x < 0. %3D

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Q: 4. Let z = -4 + 3i and zw = -7– 2i. By showing clear steps, find w in the form of a+bi.

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Q: Suppose the conjugate z is equal to the reciprocal 1/ z. What are all possible z's?

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Q: This is the solution of the Cauchy - Euler equation xy" + 5xy' + 20y = 0 %3D O y = C1x4 cos(4 ln(x))…

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Q: 1. Use the Intermediate Value Theorem to prove that the equation 3x - 4x? = 3 is solvable on the…

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Q: Prove using the IVT. For all positive integers k, cos x = xk has a solution.

A: Intermediate value theorem: Let f be a function continuous on a closed interval a,b. Let k be number…

Q: For a Cauchy-Euler DE the roots of auxiliary are m = -3 ± 3i then Ye = x (c, cos(3 In x)+ c2 sin(3…

A: Using substitution of Cauchy Euler D.E.

Q: 2.2 Find upper bound of 2+4 (z - 3i)3 +1 where C is the circle |2 – 3i| = 4.

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Q: Prove that the equation ! + = 1 has no solutions in positive integers æ, y, z.

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Q: 21.2. Find the residue of f(2) = 1/ Cos z at all its poles.

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Q: Let z1 = 2(cos() + i sin 12 and z2 = 6(cos() + i sin). Find z1z2 and (Enter your answers in a + bi…

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Q: 6. Show that when ) < |z - 1| < 2, e - ( - 3) (2 - 1)(2 – 3) Complex Analysis n=0 (z-1)" 2+2 2(z-1)

A: We use Laurent series to prove given example.

Q: 4. following is correct? A. B. C. Let p = a + jb, b > 0 be the complex root of the denominator of…

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Q: The standard form of 3dy-xydx = 3y3e4x/3  by Bernoulli's equation is:

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Q: 1. how that { cos(Inx), sin(Inx)} is a fundamental solutions set for x²y" + xy' + y = 0 on the…

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Q: The auxiliary roots of the auxiliary equation associated with a certain HLDE are m=±3, ±3i, + 3i.…

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Q: 12. Show that the equation z + iz + 1 = 0 has neither a real root nor a purely imaginary root.

A: Note: According to bartleby we have to answer only first question please upload the question…

Q: B- Find the solution to the equation = k satisfying the conditions at ax2 1-- 30 at x 0 for all…

A: A

Q: Show that sin (2n + 1)z 2n +1 { 7/4 */4 0<まく

A: We will use complex analysis to solve the given infinite sum.

Q: 7. Find a pair of integers (x,y) such that 4021x + 2014y= 1.

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Q: 3. Find the constants bo, co, a1, b1, C₁ so that So(x)=1+ box + cox² + x³, S(x) = { S₁(x) = a₁ - 2x…

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Q: 6. Show that when 0 < |-1|< 2, (z-1)(z-3) Complex Analysis (z-1)" 2+2 =-36 z−3) ² A=0 2(z-1)

A: We use substitution and Laurent series expansion.

Q: 2.2 Find upper bound of 2+4 (z – 3i)³ + 1 where C is the circle |z – 3i| = 4.

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Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: Find all values of the complex function of z for the e^(3z)=1z=x+iy

A: Its given that e3z = 1.  Given, z=x+iyWe need to sovle for z.

Q: Let z2T3-L p=S+Si O fird all the complex mumbers %3D guien by hza

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Q: If x,y,z are real numbers satisfying the equation 25 (9x* +y° )+9z²-15(5xy+yz+ 3zx )=0 then prove…

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Q: Prove that z'e'-z = 1 has exactly two roots inside the circle |z] = 1.

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Q: Show that (cos x + isin x)n= cos nx + isin nx whenever n is a positive integer. (Here i is the…

A: We can prove given result by mathematical induction: Define a statement p(n).  Show that p(n) is…

Q: 1. If z = x + iy is a complex number. (a) Verify that v2|z| > |Re(z)| + |Im(2)| [Hint: Reduce this…

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Q: Can there exist two solutions y1, 42: [-10, 10] R to the ODE y = V1+ sin (ry) %3D with yi(-10)…

A: The given differential equation is  y'=1+sin2xy On simplifying this, y'2=1+sin2xy. Hence the…

Q: 7. Derive the harmonic conjugate of u(x, y) = x² - y2. (2 2

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Q: Theorem 4.8.1 Irrationality of V2 V2 is irrational.

A: For any integer n such that n² is even then n is also even.

Q: 5) Evaluate the real integral 27T 1 de (5+ 3 cos 0)(13+5 cos 0) by complex variable methods, i.e.,…

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Q: Question is attached, Thankyou

A: Let us take z = x + iy.

Q: For z=1+i find the product of z and its complex conjugate zZ

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Q: 7.32. Prove that for 0 sxSr, - ( (* .) cos 2r 12 (a) x(w-) cos 4x 22 cos 6z 32 6. (b) æ(r - 2) = 8 /…

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Q: r(cos0+isin 0) with 0 (exact) in (0°,360°): а. z = 12-12i Z, =-3-3i z, =-4V3+ 4i b. с. Z3 = %3D

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Q: If z = 3 – 3i, express 5/z  in the form a + bi

A: The given value of z is z=3-3i.

Q: 4. Use the first Fundamental Theorem of Calculus to find 3r 2-1 dt 12 +1 sin 2z

A: Fundamental theorem of calculus Let us consider the continuous function f(x)  on the interval [a,b]…

Q: 2.2 Find upper bound of zp (z- 3i)3 + 1 where C is the circle |z - 3i = 4. 2.3 Use the Cauchy…

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Q: For a Cauchy-Euler DE the roots of auxiliary are m = -3 + 3i then Ye = x= (c, cos(3 In x)+ c2 sin(3…

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Q: Compute z -20 where z is the complex number 6(cos 70° + i sin 70°). (i) 6²º(cos 40° + i sin 40°) 1…

A: To compute z-20 where z is the complex number 6cos 70°+i sin 70°.