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Problem 1. Complex arithmetic
(a) Find the real and imaginary part of
(b) Solve 𝐀
4 − 𝐀 = 0.
√ ù
(c) Find all possible values of 𝐀.
(d) Express cos(4 𝐀 ) in terms of cos( 𝐀 ) and sin( 𝐀 ).
(e) When does equality hold in the triangle inequality ð 𝐀 1 + 𝐀 2 ð ≤ ð 𝐀 1 ð + ð 𝐀 2 ð?
(f) Draw a picture illustrating the polar coordinates of 𝐀 and 1∕ 𝐀.
Problem 2. Functions
(a) Show that sinh( 𝐀 ) = − 𝐀 sin( 𝐀𝐀 ).
(b) Give the real and imaginary part of cos( 𝐀 ) in terms of 𝐀 and 𝐀 using regular and hyperbolic sin
and cos.
(c) Is it true that ð 𝐀
𝐀 ð = ð 𝐀 ð
ð 𝐀 ð ?
Problem 3. Mappings
𝐀 − 𝐀 (a) Show that the function 𝐀 ( 𝐀 ) = maps the upper half plane to the unit disk. 𝐀 + 𝐀
(i) Show it maps the real axis to the unit circle.
(ii) Show it maps 𝐀 to 0.
(iii) Conclude that the upper half plane is mapped to the unit disk.
(b) Show that the function 𝐀 ( 𝐀 ) = maps the unit circle to the line 𝐀 = −1∕2.
𝐀 − 1
Problem 4. Analytic functions
(a) Show that 𝐀 ( 𝐀 ) = e
𝐀 is analytic using the Cauchy Riemann equations.
(b) Show that 𝐀 ( 𝐀 ) = 𝐀 is not analytic.
(c) Give a region in the 𝐀 -plane for which 𝐀 = 𝐀
3 is a one-to-one map onto the entire 𝐀 -plane.
(d) Choose a branch of 𝐀
1∕ and a region of the 𝐀 -plane where this branch is analytic. Do this so
that the image under 𝐀
1∕ is contained in your region from part (c).
Problem 5. Line integrals
(a) Compute ∫ 𝐀
𝐀 𝐀𝐀 , where 𝐀 is the unit square.
(b) Compute ∫ 𝐀
ð 𝐀 ð
𝐀𝐀 , where 𝐀 is the unit circle.
(c) Compute ∫ 𝐀
𝐀 cos( 𝐀
2 ) 𝐀𝐀 , where 𝐀 is the unit circle.
(d) Draw the region 𝐀 − { 𝐀 + 𝐀 sin( 𝐀 ) for 𝐀 ≥ 0}. Is this region simply connected? Could you define
a branch of log on this region?
𝐀
2
(e) Compute ∫ over the circle of radius 3 with center 0. 𝐀 (^) 𝐀^4 −
e
𝐀
(f) Does
∫ 𝐀
𝐀𝐀 = 0?. Here 𝐀 is a simple closed curve.
𝐀
2
∞ 1
(g) Compute ∫
4
Problem 6.
Suppose 𝐀 ( 𝐀 ) is entire and ð 𝐀 ( 𝐀 )ð > 1 for all 𝐀. Show that 𝐀 is a constant.
Problem 7.
Suppose 𝐀 ( 𝐀 ) is analytic and ð 𝐀 ð is constant on the disk ð 𝐀 − 𝐀 0
ð ≤ 𝐀. Show that 𝐀 is constant on the
disk.
Extra problems from pset 4
Problem 8. (a) Let 𝐀 ( 𝐀 ) = e
cos( 𝐀 ) 𝐀
2
. Let 𝐀 be the disk ð 𝐀 − 5ð ≤ 2. Show that 𝐀 ( 𝐀 ) attains both its
maximum and minimum modulus in 𝐀 on the circle ð 𝐀 − 5ð = 2.
Hint: Consider 1∕ 𝐀 ( 𝐀 ).
(b) Suppose 𝐀 ( 𝐀 ) is entire. Show that if 𝐀
(4) ( 𝐀 ) is bounded in the whole plane then 𝐀 ( 𝐀 ) is a
polynomial of degree at most 4.
(c) The function 𝐀 ( 𝐀 ) = 1∕ 𝐀
2 goes to 0 as 𝐀 → ∞, but it is not constant. Does this contradict
Liouville’s theorem?
Problem 9. 𝐀
Show
∫
e
cos 𝐀 cos(sin( 𝐀 )) 𝐀𝐀 = 𝐀. Hint, consider e
𝐀 ∕ 𝐀 over the unit circle.
0
Problem 10.
(a) Suppose 𝐀 ( 𝐀 ) is analytic on a simply connected region 𝐀 and 𝐀 is a simple closed curve in 𝐀 ..
Fix 𝐀 0 in 𝐀 , but not on 𝐀. Use the Cauchy integral formulas to show that
′ ( 𝐀 ) 𝐀 ( 𝐀 ) 𝐀𝐀 = ∫ 𝐀
𝐀
0 ( 𝐀^ −^ 𝐀 0 )
2
(b) Challenge: Redo part (a), but drop the assumption that 𝐀 is simply connected.
Problem 11.
cos( 𝐀 )
(a) Compute
∫ 𝐀
𝐀𝐀 , where 𝐀 is the unit circle.
𝐀
sin( 𝐀 )
(b) Compute
∫ 𝐀
𝐀𝐀 , where 𝐀 is the unit circle.
𝐀
2
(c) Compute ∫ 𝐀
𝐀𝐀 , where 𝐀 is the circle ð 𝐀 ð = 2. 𝐀 − 1
e
𝐀
(d) Compute ∫ 𝐀
𝐀𝐀 , where 𝐀 is the circle ð 𝐀 ð = 1.
𝐀
2
2 − 1 (e) Compute
∫ 𝐀
𝐀𝐀 , where 𝐀 is the circle ð 𝐀 ð = 2.
𝐀
2
(f) Compute
∫ 𝐀
𝐀𝐀 where 𝐀 is the circle ð 𝐀 ð = 2.
𝐀
2
Problem 12.
𝐀 ( 𝐀 )
Suppose 𝐀 ( 𝐀 ) is entire and lim = 0. Show that 𝐀 ( 𝐀 ) is constant.
𝐀 →∞ (^) 𝐀
MIT OpenCourseWare
https://ocw.mit.edu
18.04 Complex Variables with Applications
Spring 2018
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