Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX VARIABLESyΓγABCDxFigure 24.16 A typical cut-plane contour for use with multivalued functionsthat have a single branch point located at the origin.located at the origin is shown in figure 24.16. Here Γ is a large circle of radius Rand γ is a small one of radius ρ, both centred on the origin. Eventually we willlet R →∞and ρ → 0.The success of the method is due to the fact that because the integrand ismultivalued, its values along the two lines AB and CD joining z = ρ to z = Rare not equal and opposite although both are related to the corresponding realintegral. Again an example provides the best explanation.◮Evaluate∫ ∞dxI =, a > 0.0 (x + a) 3 x1/2 We consider the integrand f(z) =(z + a) −3 z −1/2 and note that |zf(z)| →0onthetwocircles as ρ → 0andR →∞. Thus the two circles make no contribution to the contourintegral.The only pole of the integrand inside the contour is at z = −a (and is of order 3).To determine its residue we put z = −a + ξ and expand (noting that (−a) 1/2 equalsa 1/2 exp(iπ/2) = ia 1/2 ):1(z + a) 3 z = 11/2 ξ 3 ia 1/2 (1 − ξ/a) 1/2= 1 (1+ 1 ξiξ 3 a 1/2 2 a + 3 ξ ···)28 a + .2The residue is thus −3i/(8a 5/2 ).The residue theorem (24.61) now gives∫ ∫ ∫+ +ABΓDC∫ ( ) −3i+ =2πi .γ 8a 5/2866