© Cambridge University Press 2010 Brian J. Kirby, PhD Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY Powerpoint Slides to Accompany Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices Appendix G

© Cambridge University Press 2010 Complex functions are used for many purposes, including solutions of the 2D Laplace equation with or without conformal mapping Sinusoidal functions are represented using (complex) analytic representations App G: Complex Functions

© Cambridge University Press 2010 Complex functions are written in terms of real and imaginary parts The complex conjugate has an imaginary part that is opposite that of the complex number Complex functions have magnitude and angle App G: Complex Functions

© Cambridge University Press 2010 Addition and subtraction operation for complex numbers are straightforward Multiplication occurs term by term Division uses the complex conjugate of the denominator to create a real denominator Sec G.1.1: Arithmetic Operations

© Cambridge University Press 2010 Derivatives in the complex plane can be evaluated along the real or imaginary direction Differentiability in the complex plane implies that a function’s real and imaginary parts each satisfy the Laplace equation Sec G.2: Using Complex Variables to Combine Orthogonal Parameters

© Cambridge University Press 2010 The analytic representation of a sinusoidal parameter is the complex exponential whose real part is equal to the sinusoidal parameter If all signals are at a specific frequency, we often divide out the complex exponential and retain the phasor, which carries information about phase and magnitude only Sec G.3: Analytic Representation of Harmonic Parameters

© Cambridge University Press 2010 For linear functions, the real part of the analytic representation describes the real function (the analytic representation simplifies the math) For nonlinear function, such as the product of two functions, special relations must be used (the analytic representation does not simplify the math) Sec G.3.2: Using the Analytic Representation of Harmonic Parameters

© Cambridge University Press 2010 The Kramers-Kronig relations relate the real and imaginary parts of certain types of complex functions This can be very useful when relating the reactive and dissipative parts of a material’s response to forcing Sec G.4: Kramers-Krönig Relations

© Cambridge University Press 2010 Because many systems can be solved simply by creating any differentiable complex function, transforms are very useful Conformal mapping can transform a solution for one geometry into another geometry—any transformed solution that is still differentiable still satisfies the Laplace equation Joukouwski transforms facilitate solutions around conic sections Swartz-Christoffel transforms facilitate solutions inside polygonal domains Sec G.5: Conformal Mapping