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

Q: Calculate the integral: / 3 xe xe-2x dx

A: Given,          ∫3xe-2xdx

Q: Q1. Let f = u + iv and f(z) = z. Prove that u and v satisfy Laplace's equation.

A: I have find the U and V first then the laplacian of that.

Q: Consider a subset SC C of the complex plane defined by S= {z € C : 0< |z - 2- iS 1}. (a) Sketch this…

A: Given that:S⊆ℂ and S={zℂ:0&lt;z-2-i≤1}.we have to find:(a)sketch of S.(b)we have to write closure of…

Q: 6. Consider a complex function f(z) = (1-i)+z(1+i). Show that the function is invertible (bijective)…

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Q: What is the complex variable?

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Consider the complex function of a complex variable f(z) = (z*)², where z* is the complex conjugate…

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Q: Let f: C→C be the function f(2)= wz where @ = u + iv is a fixed complex number. (a) Write down the…

A: Given f:C→C such that f(z)=ωz where ω=u+iv is a fixed number. We have to prove following three…

Q: Verify that U(x,y) = x3y - xy3 is harmonic on the complex plane.

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Q: Find the real and imaginary parts u and v (respectively) of the given complex function f as…

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Q: Let p.v. z1/2 denote the principal branch of the complex square root. Compute p.v. 21/2 dz, z +i…

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Q: Denote by Log(2) the principal branch of the complex logarithm, and let +y be as in Exercise 4.…

A: To solve the given problem we use the  defination the residue of  pole and find the integration with…

Q: 2z3+z²+4 Sc dz, z4+4z2 C: \z – 2| = 4

A: Cauchy integral theorem

Q: 3. If C is the unit circle in the complex plane C, and f(2)= =², show that f(z) dz = 0 using two…

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Q: (1) Verify that the real-valued function u(x, y) = xª – 6æ?y? + y* + 4x – 6y + 2 - defined on the…

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Q: sketch the set S of points in the complex plane satisfying Im(z) < Re(z)

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Q: (1) z – 3 +|z + 3} =10 (2) |Arg(z)< t/ 2 Arg(z) s (3) Re <1

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Q: 2-)The complex function f(z)=u(x,y)+iv(x,y) can be differentiated at any point in the complex space.…

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Q: Find a 3 x 3 matrix A such that Az = 10 for all i in R³. A =

A: Given Ax=10x  We have to find 3×3 matrix A  Now

Q: Find the real and imaginary parts u and v (respectively) of the given complex function f as…

A: Solution :- Given :-fz=z3-6z+9Let z = x+iyfz=fx+iy=x+iy3-6x+iy+9Expand it :-x+iy3-6x+iy+9

Q: 1- DETERMINE A AND BEC FOR THAT THE FUNCTION: (f:C+a=+b -2 +az +b f(=) = IF = #1-i, 2-1+i (f(1– i)…

A: Continuity of complex function:

Q: 1 Let f (z) z - 8 Part (a) Integrate the function f (z)along the curve C the complex plane given by…

A: Using the Cauchy integral theorem on the closed contour for part(b).

Q: 3y"-Gy'+6y by = = e sec x

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Q: Consider u and v are the harmonic functions that make up the real and imaginary parts of an analytic…

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Q: 1) [sex² 4x² - x² dx 2 dx 2 ਤੇ 3) sex² (4x+²) dx 2) 4) S X dx √4-9x² using Integration by…

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Q: b) Integrate the above via 6ver the complex numbers! partial frachion deomposition X= - 6 b2-4ac X+2…

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Q: Evaluate the triple integral -6 (x – 3)³ (y + 13)² (z – 2)4 - dz dy dx 8 49 625 3 by using the…

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Q: 2 Calculate the integral: dx e" V11 + e2*

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Q: . Let complex numbers a and 1/a lies on circles (x – xo)² + (y – yo)² = r² and (x- x)² + (y – yo)² =…

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Q: Show that the function u(x, y) = 2x³ - 6xy² + x² - y²-y is harmonic. Find all functions v(x, y) such…

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Q: Evaluate 9 €₂ N d da (2-1) (2³8) ²

A: Using,  Cauchy integral theorem we can solve above integral.

Q: Subject: Elements of Set Theory Greetings,I am really having difficult time with these problems that…

A: (a)We have to prove:

Q: Complex space and hoke Hl 20J. define the operator Te=(xh)k i when T Self Let H be Hilbert we To H→H…

A: Given that the operator T is defined as T:H→H by Tx=x,hk where h,k are fixed. For every operator T…