Question

Define “Smooth Path”.in complex analysis​

Answer 1

the path is continue a mapping from statement of real axis into complex number

Answer 2

SMOOTH PATH :-

Let [a..b] be a closed real interval.

Let γ:[a..b]→C be a path in C.

That is, γ is a continuous complex-valued function from [a..b] to C.

Define the real function x:[a..b]→R by:

∀t∈[a..b]:x(t)=Re(γ(t))

Define the real function y:[a..b]→R by:

∀t∈[a..b]:y(t)=Im(γ(t))

where:

Re(γ(t)) denotes the real part of the complex number γ(t)

Im(γ(t)) denotes the imaginary part of γ(t).

Then γ is a smooth path (in C) if and only if:

(1): Both x and y are continuously differentiable

(2): For all t∈[a..b], either x′(t)≠0 or y′(t)≠0.

Closed Smooth Path

Let γ be a smooth path in C.

γ is a closed smooth path if and only if γ is a closed path.

That is, if and only if γ(a)=γ(b).

Simple Smooth Path

Let γ:[a..b]→C be a smooth path in C.

γ is a simple smooth path if and only if:

(1):γ is injective on the half-open interval [a..b)

(2):∀t∈(a..b):γ(t)≠γ(b)

That is, if t1,t2∈(a..b) with t1≠t2, then γ(a)≠γ(t1)≠γ(t2)≠γ(b).