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[Electroosmosis]
[Potential Flow]
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[Debye Layer]
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Acomplex number is written in the form = a+jb, where a and b are real numbers and j = . The real part
a of the complex number is denoted by Re[], while the imaginary part b of the complex number
is denoted by the real value Im[]. Complex numbers are denoted with undertildes throughout this
text.
The complex conjugate of is denoted by ^{*} and is found by reversing the sign of the imaginary
part:
 (G.1) 
The product of a number and its complex conjugate is purely real.
A complex number can be thought of as a position vector in a two dimensional coordinate system
( = x+jy; cf. Equation 7.10). Alternately, cylindrical coordinates (magnitude and angle) can specify position in
this plane. The magnitude or absolute value of a complex number is equivalent to the radial distance in the complex
plane and is denoted by  and given by
 (G.2) 
while the angle of a complex number is equivalent to the azimuthal coordinate of the position in the complex
plane:
 (G.3) 
where we use tan^{1} to denote the twoargument inverse tangent, whose value (in the range 02π) depends on the
sign of y and x, as well as the value of y∕x:
Given Euler’s formula (exp(jθ) = cosθ+jsinθ), we can thus equivalently write
 (G.6) 
a notation termed thepolar form of the complex number.
G.1.1 Arithmetic operations
Complex numbers can be added and subtracted by adding and subtracting the real and imaginary parts separately,
so
 (G.7) 
and
 (G.8) 
Complex numbers can be multiplied term by term, noting that j^{2} = 1:
 (G.9) 
Complex numbers are divided by first multiplying the numerator and denominator by the complex conjugate
of the denominator. The resulting real denominator divides both the real and imaginary parts of the
numerator:
 (G.10) 
G.1.2 Calculus operations
Differentiation of an analytic functionwith respect to a complex number can be evaluated in a number of ways. If the
desired result as a function of , the derivative is often taken symbolically, as is the case throughout most of
Chapter 7. When evaluating the derivative for plotting in the xyplane, though, we often want to write the derivative
with respect to x or y. We do this by evaluating the limit in the definition of the derivative along either the real axis or
the imaginary axis. This leads to
There is a critical distinction here between differentiation of complex numbers as compared to real numbers. The
definition of the derivative of a real function:
 (G.12) 
must be the same for Δx approaching zero from the positive direction or the negative direction. If these two
are not the same, we say the function is not differentiable at x. For example, x is not differentiable
at x = 0, since the derivative (Equation G.12) evaluated with positive Δx is 1, while the derivative
evaluated with negative Δx is 1. For a derivative of an analytic function with respect to a complex
variable:
 (G.13) 
For a complex function two requirements must be satisfied for differentiability. First, the derivative
must be the same regardless of whether is positive or negative. Second, the derivative must be the
same regardless of whether is real or imaginary. The second condition is satisfied if Equation G.11
holds:
 (G.14) 
This means that differentiability for analytic functions has a much grander meaning than differentiability for real
functions. For real functions, we think of a differentiable function as one that is smooth. For complex functions,
differentiability implies not just smoothness, but that there is a specific relation (Equation G.14) between the
derivatives of the function in the real and imaginary directions. If we use a complex variable to represent spatial
distances in a 2D system, as we do in Chapter 7, then the differentiability of the analytic function (a
relation between derivatives along the real axis versus the imaginary axis) implies a specific spatial
relation between the derivatives in the x and ydirections. To show this, consider the complex function
= ϕ_{v} +jψ. By taking the derivatives of along the real and complex directions, we can show
that
 (G.15) 
and
 (G.16) 
These two relations are theCauchyRiemann equations for analytic functions. Combining these two relations (by
taking partial derivatives with respect to x or y), we obtain two relations:
and
So, analytic functions are differentiable only if their real and imaginary parts each satisfy the Laplace equation. In
Chapter 7, we take advantage of this by defining in terms of only. These functions are differentiable, and thus
also satisfy the Laplace equation.
[Return to Table of Contents]
Jump To:
[Kinematics]
[Couette/Poiseuille Flow]
[Fluid Circuits]
[Mixing]
[Electrodynamics]
[Electroosmosis]
[Potential Flow]
[Stokes Flow]
[Debye Layer]
[Zeta Potential]
[Species Transport]
[Separations]
[Particle Electrophoresis]
[DNA]
[Nanofluidics]
[InducedCharge Effects]
[DEP]
[Solution Chemistry]
Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections.
This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.
This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby
here. Click here for the most recent version of the errata for the print version.
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