1. Use Cauchy – Reimann Equations to identify if it is differentiable or not, if yes then differentiate the complex equation. 1. ƒ(z) = z*

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 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: 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: 16. Use Theorem 12 to show that 2n r=0

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

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Q: Use your results from problem 6 and the fact that Vz = Vr (cos (*2kz) + i sin(*2kz)) for k =…

A: Finding roots of imaginary numbers.

Q: Q2 : Express the function r2 +r(sin0 + cos 0) = r² sin² 0 by %3D cartesian form

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Q: z is a complex variable, z=x+iy question attached as photo

A: To verify the values of the two  given definite integrals

Q: 27 de Evaluate the intergal using the residue theorem. 12 + 6 sin e

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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: 5. Find the parametric equations of the line tangent to the curve C defined by the vector equation…

A: To find the parametric equation of the given tangent to the curve. The vector equation of the curve…

Q: 47. Prove that the imaginary part of the solution of z" + z' + z = te' is a solution of y"+y' +y=t…

A: Given,                           z''+z'+z=teit                                 (1)

Q: Find all the regular singular points of 2x(x + 2)y" + y' – cy = 0. Determine the indicial equation…

A: Singular point are the values of x that make P(x) equal to zero.  Here,  Px=2xx+2The singular point…

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

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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: 8. Find the positive constant a(> 0) so that the parametric x = at, y = -t? is a solution to y =…

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Q: 7)Use Taylor theorem to estimate cos(0.23) with Jerror|< 10-2

A: Here we are to estimate cosx at x=0.23=23100 with error≤10-2,using Taylor theorem i.e, using Taylor…

Q: 1.5 3-x Q1) Evaluate the integral * dy dx using polar coordinate

A: As per policy, we are solving only Question 1, Please post multiple Questions separately. (1) We…

Q: 4) Exercie S Hst 2x+iy be a complex mumber such thet (-s(37)#0Then Re (tent3z))|c equal to: Sin Bx…

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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: 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: 5. Discuss the singularities of z°(2² – 1)(z – 2)² 1/= . f(2) sin (72) Classify which of these are…

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Q: Find all the roots of z²= - /3-j A z=2 (cos +j sin-) B z=/2 (cos (- =+ 2kn) +j sin ( 12 + 2kn)), 12…

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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: The parametric equations for x y3-4 when 0s0s2n are: X-COs X-COS y-sin x-2cos30. y-25ie30…

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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: 8. Verify that the function u(x, y) = eª sin(y) is harmonic in the region where it is defined and…

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Q: Find pairs of solutions of Laplace's equation, =0, based on these complex functions (a) ze2, (b) z*:…

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3 Substitute Z=x+iy in Ze2z…

Q: If the integration by parts was used to integrate the product of e^ax and sin(bx) and cos(bx).…

A: 1. To evaluate the integral ∫eaxcosbxdx+i∫eaxsinbxdx:

Q: a) f(z) = 14/(z2 + 2z + 2) is a complex function, find (1/Tt) S-00° 14/(x2 + 2x + 2)dx %3D

A: Given:  f(z)=14z2+2z+2  1π∫-∞∞ 14x2+2x+2dx

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: 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: a) x 3cost-3, y3sint+1,0si<2a x = 3 (ost-3 y = 3smt+| 3 (os (6)-3-0 351n(0)+/= (x,y) 8TY T/4 T/2…

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Q: 7. Suppose that x2 + y? = r² and x = r cos 0, as in polar coordi- nates. Find дr ax дr and дх

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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: 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: 19.4. (a) Show that, when z is represented by polar coordinates, the C-R conditions on a function…

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Q: 27 de Evaluate the intergal using the residue theorem. 13 +4 sin 0

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Q: 5. Find the general solutions of the equations i) Z - 4z, - 5z, +10z, = cos(x+3y)-sin(x+3y) %3D ii)…

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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 prduct of Z1Z2 and the quotient of Z1/Z2. Express your answers in polar form.…

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Q: If Z1= −4+j8  ,  Z2= 8−j5  ,  Z3= 3−j7   and   Z4= −8−j9   determine ln(Z2) in both Polar and…

A: Given Z1= −4+j8  ,  Z2= 8−j5  ,  Z3= 3−j7   and   Z4= −8−j9

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.…