General Form of Cauchy's Theorem

Question 1

Define winding number.

Answer 1

Question 2

Finish the following theorem.

Answer 2

Question 3

Finish the following lemma about winding numbers.

Answer 3

Question 4

Prove the following lemma.

Answer 4

Question 5

Finish the following proposition.

Answer 5

Question 6

Prove the following proposition.

Answer 6

Question 7

Define homologous to zero.

Answer 7

Question 8

Define simply connected.

Answer 8

Question 9

Define a cycle.

Answer 9

A cycle is a formal sum of closed contours Γ = Ɣ₁ + Ɣ₂ + ….. + Ɣn

Question 10

Define the winding number of Γ around w.

Answer 10

Question 11

Define the line intergral of f over Γ.

Answer 11

Question 12

When is a cycle Γ homologous to zero in U?

Answer 12

If for every w ∉ U we have that I(Γ;w) = 0.

Question 13

What is the General Form of Cauchy’s Integral Formula, and Cauchy’s Theorem?

Answer 13

Question 14

How do you obtain the old CIF from the generalsied one?

Answer 14

Question 15

Define simple.

Answer 15

Question 16

What is Jordan’s curve theorem?

Answer 16

Question 17

Given a simple closed contour Ɣ it is possible to put an orientation on Ɣ such that for all w ∈ ℂ \ Ɣ we have that?

Answer 17

Question 18

Define when f if holomorphic on D_Ɣ^int ∪ Ɣ.

Answer 18

Question 19

What is Cauchy’s Integral Formula and Cauchy’s theorem for simple closed curves?

Answer 19