Verify that the Cauchy-Riemann equations are satisfied for the following functions. Where z is a complex variable z=x+iy  a) ez^2 b) cos(2z) c) sinh(4z)

Q: 8. If r = sin (ƒ(t)), ƒ(0) = pai/3, and ƒ′(0) = 4, then what is dr/dtat t = 0?

A: Chain Rule d(f(g(x))dx   =   f'(g(x)) . g'(x) We have given r = sin(f(t))        f(0) =  π3  ,…

Q: Is the function eiz² holomorphic in C? Explain why. What are the Cauchy-Riemann equations for eiz?

A: Given function : f(z) = ei z2

Q: then show that u and v satisfy the Cauchy-Riemann equation everywhere for ( i) and for all z # 0 in…

A:

Q: From the Laplace Equation show that there exist solutions in the form X(x) +Y(y) + Z(z)

A:

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: Compute the Jacobian of : O(r, 0) = (9r cos 0, 4r sin 0)

A:

Q: 25. Find x, y, e R from 2coshy - 7sinhx = 3 and coshy – 3 sinh? = 2 %3D %3D

A: Find x and y.

Q: Find the cubic root of z = v27+ 3i in the form of r cis 0. (Hint: tan- = 6.

A:

Q: Suppose that a particle following the path (e' cos(t). sin(), ) flies off on a tangent at lo = 2.…

A:

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: 5. Find if y = sin(vT +1). dr %3D

A:

Q: Each of the two functions v and u with the variables x and y satisfy he Cauchy - Riemann equations,…

A: Let,  u=x2-y2, v=2xy

Q: / Does the following equation satisfied Cauchy-Riemann's equation. (w=z

A:

Q: Find the tangent plane to the equation 3y cos(4x – 2y) at the point (2,4,12) Z =

A: Z=3ycos(4x-2y)  Point= (2,4,12)

Q: as Let ĝ +sinh (z?) - ji = 27 where w=f (x,y,2)- %3D Find 2 nd JW and au

A: In the given question we have to find the partial derivatives of with respect to x ,y and z. here we…

Q: Express 1+ cos a + i sin a in the modulus amplitude form.

A:

Q: Find the residues of sin(2²) at z = 0 and z = π. (sin z)2

A: To find the residue of sinz2sinz2 at z=0 and z=π Let  fz=sinz2sinz2=sinz2csc2z…

Q: Find the length of the hypocycloid defined by the equations x = a cos° (0), y = a sin° (0), 0 < 0 <…

A: Here we have to find the length of the hypocycloid defined by the equation x=acos3(θ) , y=asin3(θ)

Q: has at least one solu Prove that the equation r+ sin(x) = VI +3 %3D (0, т).

A: Here we have to prove that,  x+sinx=1x+3⇒f(x)=1x+3-x-sinx has at least one solution in the interval…

Q: Find the tangent plane to the equation 5y cos(2x – 4y) at the point (4, 2, 10). - = Z z =

A:

Q: The Cauchy-Riemann equations for f(z) is

A:

Q: (b) Show that U (x, y) = eu cos v, V(x, y) = eusin v are conjugate harmonic of each other if f = u +…

A: It is given that,  f = u+iv  is analytic. We have to show that,  U(x, y) = eu cos v  and  V(x, y) =…

Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann equations if u, = v, and u,…

A:

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

A:

Q: find 4,V for the following functions and then check the Correctness of the cauchy Riemann equations.…

A: Let fz=ux,y+ivx,y be an analytic function. The Cauchy-Riemann equations are ∂u∂x=∂v∂y∂u∂y=-∂v∂x…

Q: Show that the Euler-Cauchy Equation ry" + ary + by = 0, r+0 can be transformed into the constant…

A:

Q: Show that r = cos20 is symmetric about the y-axis.

A: I am going to solve the given problem by using some simple calculus to get the required result.

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: Find the points on the cardioid r = 4(1 – sin0) at which there is a horizontal tangent line, a…

A: According to the given information, it is required to determine the horizontal tangent line,…

Q: Let r(t) (sin(4t), cos(41), sin(4f) cos(81)). Find the point where r(t) intersects the xy-plane on…

A:

Q: a. Show that If z=x+iy where y >0 Then: 0 < |e*|<1 b. Evaluate cos (i z) – sinh?(z)

A: Using De Moivre's Theorem

Q: Calculate the length of the curve r=a*sin(^3)(x/3) for (x(1)=0, x(2)=3pi)

A:

Q: Show that r = cos(2theta) is symmetric about the y-axis.

A:

Q: iven that use the bhi tzS to compute %3D irst shifting Theorem [e"e cos 7+]

A:

Q: find directly the Laplace trans- forms of the functions described (by formula or graph) in Prob-…

A:

Q: Show that U (x, y) = e"cos v,V(x, y) = e"sin v are conjugate harmonic of each other if f = u + iv is…

A: Solution given below

Q: Use the convolution theorem to show that cos T sin (t – T) dr 1 sin t. %3D 2

A: Convolution theorem use integration of product of 2 functions and transforms them to pointwise…

Q: Verify if zo = 0 is a removable singularity for sinh z (ii) sin z3 (i) z2 (ii) 2³e/=, Explanations…

A: A limit is a mathematical object perceived to represent the last object in a sequence of the same…

Q: Evaluate the secondary 5th root of z = 50 (cos150° + j sin150°) and simplify to rectangular form O…

A:

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

A:

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: 2. Prove that, V{(sec 0 - 1)/(sec 0 + 1)} cot 0. = cosec e

A:

Q: Prove that d/dz sin(z) = cos(z) using Cauchy-Riemann equations

A: Cauchy-Riemann Equation- If f(z)=u(x, y)+i v(x,  y) is analytic, then In particular, These…

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: Let z=x+iy be a complex number such that coshz =0. Then Im(tanhz) is equal to * sinycosy…

A: first option is correct answer. Thankyou