1. Prove that |z2 = 1 2 Re(z) by: 20. The complex number z satisfies the equation |z (a) letting z = x + iy, (b) squaring the equation and then using the result z2= zz.

Q: Suppose z = x + 2i and u= 3+ yi where x, y E R. Find the smallest positive value of x z + u for…

A: Substituting z=x+2i and u=3+yi in z+uz-u:  z+uz-u=x+2i+3+yix+2i-3+yi           =x+3+2+yix-3+2-yi

Q: 2. Solve the following hyperbolic equations using suitable hyperbolic identity. a) 3cosh x + sinhx =…

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Q: Consider A = {z E C : |2| < 1}. The complex line integral eiz dz laa z(z – 2) equals: (A) 0 (В) —ті…

A: Here we can use Cauchy integral formula to calculate the integral which states that if C is any…

Q: Show that the equation 2²2 2 2²y Əy +2k c². əx² de ² Ət possesses solutions of the form 0 - kt Σ Cre…

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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: 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: 1. Show that each of the functions on the left is a 3-parameter family of solutions of the DE on its…

A: Solve differential equation.

Q: 3. Use the theorem in Sec. 48 to show that (z – zo)"- dz = 0 (n = ±1, ±2, ...) when Co is any closed…

A: The given question is solved below.

Q: 27 1 1. The integral de can be converted into the following complex integral using a suitable…

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Q: The Imaginary part of the complex function f(z) is, v(x, y) = x³ – 3x²y + 2x + 1 + y³ – 3xy² Prove…

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Q: Q1/Prove that the polar expression for the function y = Vx2 - (x2 + y2)3/2 is r = cos20

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Q: please solve any of them

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Q: d the product z,z2 and the quotient 4. Express your answers in polar form. Z2 24-(co0(플) + 1sin(즉),…

A: Here we have,  z1=3cosπ3+isinπ3, z2=8cosπ6+isinπ6

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: Find the cubic root of z = V27 + 3i in the form of r cis 0. 1 (Hint: tan =

A: Given:- Z=√27+3i To find:-  Cubic root of given complex number

Q: 4. Find the distinct roots of z for the following equation, z4 = 3i. %3D 5. Prove by taking z = a +…

A: Q4) Given that z4=3i So z4=3iz4=3cosπ2+isinπ2z4=3cos2nπ+π2+isin2nπ+π2         Since sin and cos are…

Q: 4.1 z + cos(72) -dz, c z(z² +1) with C : |z – i| =.

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Q: 1. Let n e N. Find all complex solutions to the equation: z" = i.

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Q: Q1/Prove that the polar expression for the function y = Vx2 - (x2 +y2)3/2 is r = cos20

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

Q: Example 1 : Obtain the complex integral: zdz where C is the straight line path from z = 1+i to z = 3…

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Q: Q3/ Find the complex integral 3+i z2 dz 0+0i Along the path 1 y ==x 3

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Q: 18 In(z – 3) Question 10) Find I = -dz, where C = Iz - 2 + 4i| = 4, Ccw z2 + (4 + 4i)z + 3+ 4i

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Q: 6. Find the general solutions of X' = AX with A = 5 4 Use X = eat [c1{Reß cos bt – Im ß sin bt}+…

A: According to the given information, it is required to find the general solution of X’ = AX.

Q: Q1:-/ Verify by direct substitution that y, = epx cos qx and y2 = ePx sin qx , are solutions of the…

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Q: 10 Consider A = {z € C : |2|< 1}. The complex line integral Jonate=2) dz equals: (A) 0 (B)-ni+e²/2…

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Q: Solve the problem -64u u(0, t)-u(1, t)-0 IC u(a, 0)- 5 sin(2xz), PDE 00 BC 4,(z, 0) - 3 sin(3rz)…

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Q: By Cauchy's integral formula, Taylor's Theorem, or any method under complex analysis (specify…

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Q: a) f(z) = 14/(z2 + 2z + 2) is a complex function, find (1/Tt) S-00° 14/(x²+ 2x + 2)dx

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Q: 1. Solve the following Cauchy-Euler equation. x*y" – 4xy' + 6y = 2x* + x?

A: To solve the below Cauchy-Euler equation. x2y''−4xy'+6y=2x4+x2

Q: 9. Suppose that z(t) = x (t) +iy(t) is a solution of z" + pz' + qz = Ae'o". (5.31) Substitute z(t)…

A: Given - Suppose that zt = xt + iyt is a solution of z'' + pz' + qz = Aeiωt            5.31 To find -…

Q: do Evaluate the intergal using the residue theorem. 12 +1 cos 0

A: ∫02πdθ12+1 cosθLet, z=eiθ . Then cosθ=(z+z-1)/2, dz=iz dθ, so that∫02πdθ12+1…

Q: 1.Why does the following DE have an unique solution on an open rectangle that contains the point (O;…

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Q: Determine whether y = e', y, = sin2t, ya= cos2t make up a fundamental set of solutions by finding…

A: The Wronskian matrix is constructed using y1 , y2 , y3 and their derivatives. If the determinant of…

Q: 2 de Evaluate the intergal using the residue theorem. 12 + 6 sin 0

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Q: Verify that the Cauchy-Riemann equations are satisfied for the following functions. Where z is a…

A: To verify the Cauchy -Riemann equations for the given analytic functions f(z)

Q: Find all the regular singular points of (4 – x2)y" + 2xy' + 3y = 0 by using the definition.…

A: The given question is solved below.

Q: Evaluate the complex integral e19.2.png rx+yi | Vaz + b dz if a = 2, b = 14, c = 16, x = 2, and y =…

A: The given problem is to evaluate and simplify the given complex integral with unknown variables…

Q: 1. Show that To = -1 is a regular singular point of the equation (1– 2)y" – 2ry + a(a+ 1)y = 0,…

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Q: • Find a harmonic conjugate of u(r, y) = x² – 2x – y². Show that u(r, y) = e" cos y is harmonic.

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Q: Find the general solution to the hypergeometric ODE #(1 – 2)y" + (; - 2-) a - - 0. 3 ) y y = 0.

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Q: lim,-9+76i (z² + 18z) = x + iy,

A: The solution is given as

Q: (10) The result of (D - 3D,D, + 2D )sin(x – 2y) is (a) -15cos(x-2y) (b) -15sin(x-2y) (c)…

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Q: 4. Verify that the functions 1, r, cos r and sinr form a funda- mental set of solutions of y(4) + y"…

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Q: Evaluate the complex integral mq3.1.png •x+yi | asinh(bz) dz if a = 7.4, b = 3,v = 5, x = 3 and y =…

A: The given problem is to evaluate the given complex integral with unknown variables values and also…

Q: 27 1 1. The integral de can be converted into the following complex integral using a suitable…

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Q: Determine whether Y = e', y2 = sin2t, y3 = cos2t make up a fundamental set of solutions by finding…

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Q: Prove that: Z,Z2 =r,r2[cos(0, + 02) + i sin(0, + 02)] 1.

A: NOTE- As we are allowed to answer one question in one session so, i have answered first one for you.…

Q: Q3/ Find the complex integral 3+1 * z? dz Along the path y =3*

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Q: Find the solution of the following Cauchy problem and sketch its graph att = 0,0.5, 1, 1.5, 2 & 2.5:…

A: Consider the given utt=uxx,              -∞&lt;x&lt;∞, t&gt;0u(x,0)=ϕ(x)=2cosπx4 ,|x|≤20…

Q: Evaluate the complex integral e19.2.png x+yi Vaz + b dz 0. if a = 10, b = 19, c = 10, x = 1, and y =…

A: To find- Evaluate the complex integral ∫0x+yiaz + b dz if a = 10, b = 19, c = 10, x = 1, and y = 5.…