Ch.4:Complex Integration Ch.4:Complex Integration L4.1 Contours L4.1 Contours LCurves Directed Smooth Arcs Concept of a Contour A smooth arc,together with a specific ordering of its points, The general curves are formed by joining directed smooth is called a directed smooth arc.The ordering can be curves together end-to-end;this allows self-intersection,cusps, indicated by an arrow and corners The point z(t1)will precede z(t2)whenever t1<t2.Since It will be convenient to include single isolated points as there are only two possible ordering,any admissible members of this class parametrization must fall into one the two categories, Definition according to the particular ordering it respects A contour I is either a single point zo or a finite sequence of If z=z(t),a<t<b,is an admissible parametrization directed smooth curves (1,72,...,Yn)such that the terminal consistent with one of the ordering,then point of y coincides with the initial point of+for each z =z(-t),-b<t <-a,always corresponds to the opposite =1,2,...,n-1.In this case one can write ordering T=m+2+.+m Ch.4:Complex Integration Ch.4:Complex lategration L4.1 Contours 4.1 Contours LCurves Contours Directed Smooth Arcs(Cont'd) Concept of a Contour(Cont'd) The theory of contour is easier to express in terms of contour The points of a smooth closed curve have been ordered when parameterizations (i)a designation of the initial point is made and (ii)one of the two "directions of transit"from this point is selected One can say that z=z(t),a<t<b,is a parametrization of the contour I=(1,72,...,n)if there is a subdivision of If this parametrization is given by z=z(t),a <t <b,then (i) [a,b]into n subintervals [ro,],[1,T2],...,[Tn-1,Tn],where the initial point must be z(a)and (ii)the point z(t1)precedes a=TO<TI<...Tn-1 Tn =b,such that on each the point z(t2)whenever a<ti<t2<b subinterval 1,T]the function z(t)is an admissible The phrase directed smooth curve will be used to mean either parametrization of the smooth curve y,consistent with the a directed smooth arc or a directed smooth closed curve direction on Next,we are ready to specify the more general kinds of curves Since the endpoints of consecutive 's are properly that will be used in the theory of integration connected,z(t)must be continuous on [a,b].However z'(t) may have jump discontinuities at the points yCh.4: Complex Integration 4.1 Contours Curves Directed Smooth Arcs A smooth arc, together with a specific ordering of its points, is called a directed smooth arc. The ordering can be indicated by an arrow The point z(t1) will precede z(t2) whenever t1 < t2. Since there are only two possible ordering, any admissible parametrization must fall into one the two categories, according to the particular ordering it respects If z = z(t), a ≤ t ≤ b, is an admissible parametrization consistent with one of the ordering, then z = z(−t), −b ≤ t ≤ −a, always corresponds to the opposite ordering Ch.4: Complex Integration 4.1 Contours Curves Directed Smooth Arcs (Cont’d) The points of a smooth closed curve have been ordered when (i) a designation of the initial point is made and (ii) one of the two ”directions of transit” from this point is selected If this parametrization is given by z = z(t), a ≤ t ≤ b, then (i) the initial point must be z(a) and (ii) the point z(t1) precedes the point z(t2) whenever a<t1 < t2 < b The phrase directed smooth curve will be used to mean either a directed smooth arc or a directed smooth closed curve Next, we are ready to specify the more general kinds of curves that will be used in the theory of integration Ch.4: Complex Integration 4.1 Contours Contours Concept of a Contour The general curves are formed by joining directed smooth curves together end-to-end; this allows self-intersection, cusps, and corners It will be convenient to include single isolated points as members of this class Definition A contour Γ is either a single point z0 or a finite sequence of directed smooth curves (γ1, γ2,...,γn) such that the terminal point of γk coincides with the initial point of γk+1 for each k = 1, 2,...,n − 1. In this case one can write Γ = γ1 + γ2 + ... + γn Ch.4: Complex Integration 4.1 Contours Contours Concept of a Contour (Cont’d) The theory of contour is easier to express in terms of contour parameterizations One can say that z = z(t), a ≤ t ≤ b, is a parametrization of the contour Γ=(γ1, γ2,...,γn) if there is a subdivision of [a, b] into n subintervals [τ0, τ1], [τ1, τ2],..., [τn−1, τn], where a = τ0 < τ1 <...< τn−1 < τn = b, such that on each subinterval [τk−1, τk] the function z(t) is an admissible parametrization of the smooth curve γk, consistent with the direction on γk Since the endpoints of consecutive γk’s are properly connected, z(t) must be continuous on [a, b]. However z(t) may have jump discontinuities at the points γk