M417

1.

Find

dtan−1 z dz

2.

a)

Find a [multivalued] function F₊(z) such that [(dF₊)/dz] = [1/(1 + i z)].

b)

Find a [multivalued] function F₋(z) such that [(dF₋)/dz] = [1/(1 − i z)].

3.

Show that

1 1 + z2 = 1 2 ⎧ ⎨ ⎩ 1 1 + i z + 1 1 − i z ⎫ ⎬ ⎭ .

4.

Find a (single valued!) function F(z) defined in the strip

{−1 < ℑz < 1 }

such that

•

F(z) is analytic (single valued!),

•

[dF /dz] = [1 /(1 + z²)],

•

F(0) = π.