Dot and cross products. Complex conjugate coordinates. Complex differential operators. Gradient, divergence, curl and Laplacian of complex functions.

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Vector interpretation of complex numbers. A complex number z = x + iy can be viewed as a vector whose initial point is the origin O and whose terminal point P is point (x, y) as shown in Fig. 1. Two vectors having the same length and direction but different initial points, such as OP and AB in Fig. 1, are considered equal. Thus a single complex number can represent any of many vectors — any vector in the plane of the same length and direction — as is the usual case with vectors.

Addition of complex numbers corresponds to the parallelogram law for the addition of vectors. Thus the sum of z₁ and z₂ corresponds to the diagonal OB of the parallelogram shown in Fig. 2.

Dot product of two complex numbers. The dot product of the two complex numbers z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ is defined as

1) z₁∘z₂ = |z₁||z₂|cos θ = x₁x₂ + y₁y₂

where θ is the angle between z₁ and z₂ and assumed to be less than 180^o.

Syn. scalar product

Cross product of two complex numbers. The cross product of the two complex numbers z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ is defined as

2) z₁×z₂ = |z₁||z₂|sin θ = x₁y₂- y₁x₂

where θ is the angle between z₁ and z₂ and assumed to be less than 180^o.

From 1) and 2), remembering that e^{i θ} = cos θ + i sin θ, we can easily derive the following theorem:

Theorem.

= z₁∘z₂ + i(z₁×z₂) = |z₁||z₂| e^{i θ}

If z₁ and z₂ are non-zero, then

1. A necessary and sufficient condition that z₁ and z₂ be perpendicular is that z₁ z₂ = 0.

2. A necessary and sufficient condition that z₁ and z₂ be parallel is that z₁ z₂ = 0.

3. The magnitude of the projection of z₁ on z₂ is |z₁ z₂|/|z₂| .

4. The area of a parallelogram having sides z₁ and z₂ is |z₁ z₂| .

Spiegel. Complex Variables. p. 6

Complex conjugate coordinates. The location of points in the complex plane can be specified by rectangular coordinates (x, y) or polar coordinates (r, θ). We shall now introduce another set of coordinates by which it is sometimes convenient to identify points in the complex plane. It is the number pair where is the conjugate of z. They are called the complex conjugate coordinates or briefly the conjugate coordinates of the point. The transformation equations relating the conjugate coordinates with the rectangular coordinates are

and

Sometimes, for one reason or another, one may wish to use some non-rectangular coordinates, such as cylindrical, spherical, curvilinear, etc. on some problem instead of the usual rectangular coordinates. Another system may be more natural and convenient. We can do this if we wish. It is only necessary that there exist a one-to-one between the coordinates of the two systems. Conjugate coordinates represent a kind of curvilinear coordinates. But conjugate coordinates seem very awkward, obscure and useless. Why would we have any interest in conjugate coordinates? The answer is they are often very useful in computing things like the gradient, divergence, curl and Laplacian of a function. To understand why first note that the functions presented to us are often elementary functions (functions containing algebraic, trigonometric, logarithmic, etc. terms, functions like 3z² + sin z + 1) in the complex variable z. To compute things like the gradient, divergence, curl, etc. we could substitute z = x + iy into the function and expand, thus changing it into a function in x and y, and then compute the gradient, divergence, curl, etc. of that function in the usual way. However, that way is tedious and there is an easier way. We can, in fact, compute the gradient, divergence, curl, etc. directly from the function in the z form. In fact, the function in z form is already expressed in complex conjugate coordinates! It is a function in the variable z only but it is actually a function in z and . It is just that the is absent. For insight into this consider a couple of things. First note that any elementary function in the complex variable z is analytic. Now consider the following theorem:

Theorem 1. If in any analytic function w = u(x, y) + i v(x, y) the variables x and y are replaced by their equivalents in terms of z and , namely,

w will appear as a function of z alone.

Proof

If we were given a function in the variable z and substituted z = x + iy into it to change it to a function in x and y and then substituted

into it, the new expression would initially contain the variable , but the would fall out and we would get back our original expression in the variable z alone. And Theorem 1 tells us why the falls out. The expression is analytic.

Complex differential operators. We define the differential operators ∇ (del) and (del bar) as follows:

Derivation for conjugate coordinates

Gradient, divergence, curl and Laplacian

We will use the operator ∇ to define the gradient, divergence, curl and Laplacian of various functions. In the discussion we will use the following notation:

● F(x, y) represents a real continuously differentiable function of x and y (a scalar)

● A(x, y) = P(x, y) + i Q(x, y) represents a complex continuously differentiable function of x and y (a vector)

● G(z, ) represents the function obtained by substituting

in F(x, y).

● B(z, ) represents the function obtained by substituting

in A(x, y).

Gradient of a real function. We define the gradient of a real function F (scalar) by

Geometrically, this represents a vector normal to the curve F(x, y) = c where c is a constant.

Gradient of a complex function. We define the gradient of a complex function A = P + iQ (vector) by

We note that if B is an analytic function of z, then . Thus the gradient is zero, i.e.

showing the Cauchy-Riemann equations are satisfied in this case.

Divergence of a complex function. By using the dot product of two complex numbers as extended to operators, we define the divergence of a complex function (vector) by

The divergence is always a real function.

Curl of a complex function. By using the cross product of two complex numbers as extended to operators, we define the curl of a complex function (vector) by