Changing the integration order

1 Analysis: Questions

1. (a) Calculate
   Z 1
   0
   Z 1
   0
   sin (e
   x
   ) dx
   dy +
   Z e
   1
   Z 1
   ln y
   sin (e
   x
   ) dx
   dy.
   Tip: Drawing the integration area and changing the integration order can be
   an opportunity to attack the problem.
   (b) Calculate the volume of the body
   K =
   
   (x, y, z) : x
   2 + y
   2 + z
   2 ≤ 12, z ≤ 1, x + y > 0
2. (a) Formulate Stokes’s theorem with all predictions.
   (b) Verify your Stoke’s theorem for the vector field F(x, y, z) = (z
   2
   , x2
   , y2
   )
   and the area Y given by
   (x − z)
   2 + (y + z)
   2 = z
   2
   for 0 ≤ z ≤ 1.
   That is, calculate the curve integral R
   (∂ Y ) F.dr both directly and with the help of
   Stokes’s theorem. Your solution should clearly state how you choose orientation.
3. Betrakta
   Z
   Γ
   x + y
   x
   2 + y
   2
   dx +
   −x + y
   x
   2 + y
   2
   dy.
   (a) Calculate the curve integral, where the curve Γ goes in the half-plane y ≥ 0
   from the point (1, 0) to (−1, 0) along the superellipse x
   6 + 3y
   6 = 1.
   (b) List all the statements you have used in (a) and explain how.
   Note: If in the formulation of the statement, e.g. a field F, then indicate
   how you have chosen F in the example,and explain why all the set of prerequisites are met.
   (c) Enter a curve Γ such that the curve integral becomes zero and justify your
   choice.
4. (a) Consider the series
   X∞
   n=0
   
   2n
   n
   
   z
   n
   . (1)
   i. Determine the radius of convergence of the series.
   ii. Enter an amount of M ⊂ C such that the series converges uniformly in
   M.
   iii. The new concepts of uniform convergent function sequence and uniform
   convergent function series. Explain the difference between point-by-point and
   uniform convergence pursuits.
   (b) Consider the series
   X∞
   n=1
   (−1)n
   n · 9
   n
   z
   2n
   .
   1
   i. Determine the radius of convergence of the series R.
   ii. Determine the value of the series for |z| < R.
   iii. Does the series also converge for all z C with |z| = R?
   Motivate your answers!
5. Let f be a complex value function in an open subset of the complex speech
   plane C.
   (a) Enter the definition of an analytical function.
   (b) State two properties of f that are equivalent to the analytical property.
   (c) Determine all real-value analytical functions f in C, that is, determine all
   analytical functions f(z) = u(x, y) + iv(x, y) with imaginary part v(x; y) = 0
   for all x and y (where as usual z = x + iy).
6. Let f be a function that is analytic in an open quantity containing the
   closed unit circle disk D = {z C; |z| ≤ 1} .
   Show: If |f (x)| ≤ 1 for all z with |z| = 1, then |f (0)| ≤ 1 also applies.
   Tip: Cauchy’s integral formula and the triangle probability of integrals can be
   helpful.
   

Sample Solution