Reference no: EM13711579

Question:

This is a complex analysis assignment

(i) The theorem stated below is followed by statements from its proof. For each of the numbered statements provide a short comment that explains or justifies it.

Cauchy's theorem for a triangle. Let f be a function that is holomorphic in a simply connected region R and let T be a triangle in R with boundary ∂T. Then

∫_∂T f = 0.

Proof There exists a nested sequence of triangles {T_n} such that

(1/4)ⁿ|∫_∂T f| ≤ |∫_∂Tn f| and l(∂T_n) = (1/2ⁿ) l(∂T),

and there exists z_o  such that zo ∈ T_n for all n.

(1) Given any ∈ > 0 there exists ∂ > 0 such that

|z - z_o| < δ => |f(z) - f(z₀) - (z - z₀)f'(z₀)|≤ ∈|(z) - (z_o)|.

(2) There exists n such that l(∂T_n) < δ and T_n ⊂ D(z_o:δ)

(3) For z ∈ ∂T_n

|fz - f(z_o) - f'(z_o)(z - (z_o)| ∈l()< ∂T_n.

(4) ∫_∂Tn f(z)dz = ∫_∂Tn (f(z) - f(z_o) - f'(z_o)(z - (z_o)dz.

(5) |∫_∂Tn f| ≤ ∈l(∂T_n)l(∂T_n)

(6) (1/4)n|∫_∂T f| ≤ ∈l(∂T_n)l(∂T_n) ≤ ∈(1/4)ⁿl(∂T)ll(∂T),

(7) ∫_∂T f = 0.

(ii) Let R be any rectangular contour in C and let f be a function that is holomorphic in C. Use the theorem above to prove that

∫_R f = 0

where R is any rectangular contour in C.