Residue theorem application

One of the many applications of the theory of complex variables is the application of the residue theorem to evaluate definite real integrals. Another is to use conformal mapping to solve boundary-value problems involving harmonic functions. The residue theorem is also very useful in evaluating integrals resulting from solutions of differential equations by the method of integral transforms. 

Actually, for the concerned function two poles are identified in the complex plane of Figure 3.13, producing the associated integrals to be solved according with the Cauchy residues theorem with application to the single or to multiple poles as well ... 

The third chapter addresses linear second-order ordinary differential equations. A brief discourse, it reviews elementary differential equations, and the chapter serves as an important basis to the solution techniques of partial differential equations discussed in Chapter 6. An applications section is also included with ten worked-out examples covering heat transfer, fluid flow, and simultaneous diffusion and chemical reaction. In addition, the residue theorem as an alternative method for Laplace transform inversion is introduced. 

By studying how Q, transforms under a rigid rotation of the capillary charge Il(r), one can show easily that the complex-valued multipole charge can be written as gj = Inserting this into Equation 2.37 one arrives at Equation 2.12. A definition of the multipole charge alternative to Equation 2.38 is provided by application of the residue theorem to Equation 2.37 ... 

After the application of Green s theorem to the second order term in Equation (2.81) we get the weak form of the residual statement as... 

Numerous empirical correlations for the prediction of residual NAPL dissolution have been presented in the literature and have been compiled by Khachikian and Harmon [68]. On the other hand, just a few correlations for the rate of interface mass transfer from single-component NAPL pools in saturated, homogeneous porous media have been established, and they are based on numerically determined mass transfer coefficients [69, 70]. These correlations relate a dimensionless mass transfer coefficient, i.e., Sherwood number, to appropriate Peclet numbers, as dictated by dimensional analysis with application of the Buckingham Pi theorem [71,72], and they have been developed under the assumption that the thickness of the concentration boundary layer originating from a dissolving NAPL pool is mainly controlled by the contact time of groundwater with the NAPL-water interface that is directly affected by the interstitial groundwater velocity, hydrodynamic dispersion, and pool size. For uniform... 

The application of the residue (Cauchy) theorem leaves with ... 

---