Cauchy Principal Value Integrals柯西主值积分.pptVIP

Review of Cauchy Principal Value Integrals Cauchy Principal Value Integrals Brief Review of Singular Integrals Singular Integrals, cont’d Singular Integrals, cont’d Singular Integrals, cont’d Dispersion Relations Dispersion Relation, Example 1 Dispersion Relation, Example 2 Integrals of the form  Integrals of the form (cont.)  Integrals of the form Integrals of the form (cont.) Integrals of the form (Fourier Integrals) Integrals of the form (Fourier Integrals) (cont.) Exponential Integrals Exponential Integrals (cont.) Exponential Integrals (cont.) Exponential Integrals (cont.) Integration around a Branch Cut Integration around a Branch Cut (cont.) Integration around a Branch Cut (cont.) Integration around a Branch Cut (cont.) * D. R. Wilton ECE Dept. Fall 2009 ECE 6382 Evaluation of Definite Integrals Via the Residue Theorem Recall for real integrals, but a finite result is obtained if the integral interpreted as because the infinite contributions from the two symmetrical shaded parts shown exactly cancel. Integrals evaluated in this way are said to be (Cauchy) principal value integrals (or “deleted” integrals) and are often written as x 1/x To evaluate consider the integral Note: Principal value integrals have either symmetric limits extending to infinity or a vanishing, symmetric deleted interval at a singularity. Both types appear in this problem! Logarithmic singularities are examples of integrable singularities: 1 1/x singularities are examples of singularities integrable only in the principal value (PV) sense. Principal value integrals must not start or end at the singularity, but must pass through to permit cancellation of infinities Singularities like 1/x2 are non- integr