(5.4.2) This is cute proof that the Pythagorean identity
extends to the complex plane.
For this problem let 
. Do the steps below and then
combine them to justify 
- Show that the function g ( z ) is entire
- Compute the derivative g ′ ( z )
- Compute g ( 0 )
Show sin( z ) = sin( x )cosh( y ) + i cos( x )sinh( y )
Where is 
differentiable? Analytic?
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(5.4.2) This is cute proof that the Pythagorean identity
extends to the complex plane.
For this...
Show that the real and imaginary parts of the complex-valued function f(x) = cot z are...
Show that the real and imaginary parts of the complex-valued function f(x) = cot z are - sin 2.c sinh 2g u(I,y) v(x,y) = cos 2x - cosh 2y cos 2x - cosh 2y (cot 2 = 1/tan 2)
1. Starting with the equation a tanh y show that, for real r, y and |rl...
1. Starting with the equation a tanh y show that, for real r, y and |rl < 1, y-tanh-1 2. Compute the following derivative dtanh (sin() 3. Assume θ is a real number. Then use Euler's formula eie-cos θ + isin θ to show that coth(i0)-icot(e) 4. Use the definitions to obtain an equation for cosh(3x) in terms of cosh(x) and sinh(x) and their various products (e.g., cosh*(z), cosh(x) sinh3(x) etc.). Do not use the double-angle formula such as cosh(u+...
Show all the work 5. Compute the flux (integral) of the vector field )-(7777 규) along...
Show all the work
5. Compute the flux (integral) of the vector field )-(7777 규) along the surface Σ of exercise 4 with respect to φ 4. Let Σ be the piece of the hyperboloid x2+92-2-1 between the planes z-4/3 and z 12/5. Compute the integral of the function f(x,y,z) = z? along Σ Hint: use the parametrization (change of coordinates) given by φ(u, θ)-(cosh u cos θ, cosh u sin θ, sinh u) and remember the elementary properties of...
NOTE: Show all steps in your solutions. Only partial credit will be given if steps are...
NOTE: Show all steps in your solutions. Only partial credit will be given if steps are not shown though the final answer is correct. 1. Show that the real and imaginary parts of the complex-valued function f(2) = cot z are - sin 2x sinh 2 u(x, y) v(x,y) cos 2.c – cosh 2y' cos 2x - cosh 2y (cot z = 1/ tan ) [20 points)
Consider the following complex-variable function cosh a < T f(z) la! cosh πχ, a) Find all its singularities, state their nature and compute the residues b) Consider the rectangular contour y with...
Consider the following complex-variable function cosh a < T f(z) la! cosh πχ, a) Find all its singularities, state their nature and compute the residues b) Consider the rectangular contour y with vertices at tR and tRi. Evaluate 6 6 dz cosh πχ c) Using the previous result take the limit R-to prove that cosh ax (10] 2 cos (g Hint: remember that cosh(a + b) -cosh a cosh b + sinh a sinh b d) Why is the above...
NOTE: Show all steps in your solutions. Only partial credit will be given if steps are...
NOTE: Show all steps in your solutions. Only partial credit will be given if steps are not shown though the final answer is correct. 1. Show that the real and imaginary parts of the complex-valued function f(x) = cot z are sin 2.c sinh 2 u(x,y) = v(x,y) cos 2. - cosh 2y' cos 2. - cosh 2y (cot z = 1/tanz) [20 points) 2. Obtain the equilibrium points of the following system of 1st or- der ODE and classify...
5. Use a substitution and an integration by parts to find each of the following indef-...
5. Use a substitution and an integration by parts to find each of the following indef- inite integrals: (b) | (cos(a) sin(a) esas) de (a) / ( (32 – 7) sin(5x + 2)) de (c) / (e* cos(e=)) dt (d) dr 6. Spot the error in the following calculation: S() will use integration by parts with 1 We wish to compute dr. For this dv du 1 dar = 1. This gives us dr by parts we find dr =...
Problem 5. Prove that parametric equations: x a-cosh(s) (a > 0) or back half(a < 0) of hyperboloid of one she...
Problem 5. Prove that parametric equations: x a-cosh(s) (a > 0) or back half(a < 0) of hyperboloid of one sheet: Χ t), y b-sinh(s) cos (t) zc-sinh(s) sin( t), (x,y,z) lies on the front half L" a2 b2 c2 Problem 6 What graph of this Compute the arc length : rit)- < sin t, cos t, 2Vt', when 0<t < function: a) Compute the arc length : re)-3cos(9) and 0 < θ < π/2 b) Problem 7. Find parametric...
real analysis 1,2,3,4,8please 5.1.5a Thus iff: I→R is differentiable on n E N. is differentiable on / with g'...
real analysis
1,2,3,4,8please
5.1.5a
Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative...
2. Consider the vector field F = (yz - eyiz sinx)i + (x2 + eyiz cosz)j...
2. Consider the vector field F = (yz - eyiz sinx)i + (x2 + eyiz cosz)j + (cy + eylz cos.) k. (a) Show that F is a gradient vector field by finding a function o such that F = Vº. (b) Show that F is conservative by showing for any loop C, which is a(t) for te (a, b) satisfying a(a) = a(6), ff.dr = $. 14. dr = 0. Hint: the explicit o from (a) is not needed....
Show that the real and imaginary parts of the complex-valued function f(x) = cot z are - sin 2.c sinh 2g u(I,y) v(x,y) = cos 2x - cosh 2y cos 2x - cosh 2y (cot 2 = 1/tan 2)
1. Starting with the equation a tanh y show that, for real r, y and |rl < 1, y-tanh-1 2. Compute the following derivative dtanh (sin() 3. Assume θ is a real number. Then use Euler's formula eie-cos θ + isin θ to show that coth(i0)-icot(e) 4. Use the definitions to obtain an equation for cosh(3x) in terms of cosh(x) and sinh(x) and their various products (e.g., cosh*(z), cosh(x) sinh3(x) etc.). Do not use the double-angle formula such as cosh(u+...
Show all the work
5. Compute the flux (integral) of the vector field )-(7777 규) along the surface Σ of exercise 4 with respect to φ 4. Let Σ be the piece of the hyperboloid x2+92-2-1 between the planes z-4/3 and z 12/5. Compute the integral of the function f(x,y,z) = z? along Σ Hint: use the parametrization (change of coordinates) given by φ(u, θ)-(cosh u cos θ, cosh u sin θ, sinh u) and remember the elementary properties of...
NOTE: Show all steps in your solutions. Only partial credit will be given if steps are not shown though the final answer is correct. 1. Show that the real and imaginary parts of the complex-valued function f(2) = cot z are - sin 2x sinh 2 u(x, y) v(x,y) cos 2.c – cosh 2y' cos 2x - cosh 2y (cot z = 1/ tan ) [20 points)
Consider the following complex-variable function cosh a < T f(z) la! cosh πχ, a) Find all its singularities, state their nature and compute the residues b) Consider the rectangular contour y with vertices at tR and tRi. Evaluate 6 6 dz cosh πχ c) Using the previous result take the limit R-to prove that cosh ax (10] 2 cos (g Hint: remember that cosh(a + b) -cosh a cosh b + sinh a sinh b d) Why is the above...
NOTE: Show all steps in your solutions. Only partial credit will be given if steps are not shown though the final answer is correct. 1. Show that the real and imaginary parts of the complex-valued function f(x) = cot z are sin 2.c sinh 2 u(x,y) = v(x,y) cos 2. - cosh 2y' cos 2. - cosh 2y (cot z = 1/tanz) [20 points) 2. Obtain the equilibrium points of the following system of 1st or- der ODE and classify...
5. Use a substitution and an integration by parts to find each of the following indef- inite integrals: (b) | (cos(a) sin(a) esas) de (a) / ( (32 – 7) sin(5x + 2)) de (c) / (e* cos(e=)) dt (d) dr 6. Spot the error in the following calculation: S() will use integration by parts with 1 We wish to compute dr. For this dv du 1 dar = 1. This gives us dr by parts we find dr =...
Problem 5. Prove that parametric equations: x a-cosh(s) (a > 0) or back half(a < 0) of hyperboloid of one sheet: Χ t), y b-sinh(s) cos (t) zc-sinh(s) sin( t), (x,y,z) lies on the front half L" a2 b2 c2 Problem 6 What graph of this Compute the arc length : rit)- < sin t, cos t, 2Vt', when 0<t < function: a) Compute the arc length : re)-3cos(9) and 0 < θ < π/2 b) Problem 7. Find parametric...
real analysis
1,2,3,4,8please
5.1.5a
Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative...
2. Consider the vector field F = (yz - eyiz sinx)i + (x2 + eyiz cosz)j + (cy + eylz cos.) k. (a) Show that F is a gradient vector field by finding a function o such that F = Vº. (b) Show that F is conservative by showing for any loop C, which is a(t) for te (a, b) satisfying a(a) = a(6), ff.dr = $. 14. dr = 0. Hint: the explicit o from (a) is not needed....
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