Geometric arrangement for Fresnel's calculation Consider the case of a point source located at a point P < sub > 0 </ sub >, vibrating at a frequency f. The disturbance may be described by a complex variable U < sub > 0 </ sub > known as the complex amplitude.

In Consider Phlebas it is noted that Minds still find humans fascinating, especially their odd ability to sometimes achieve similarly advanced reasoning as their much more complex machine brains.

Consider a slightly more complex example:

Consider an open subset U of the complex plane C. Let a be an element of U, and f: U

Consider the complex Hilbert space L < sup > 2 </ sup > and the differential operator

Consider the complex Hilbert space L < sup > 2 </ sup >( R ), and the operator which multiplies a given function by x:

Consider the ( Euclidean ) complex plane equipped with the metric

Consider the Gram – Schmidt process applied to the columns of the full column rank matrix, with inner product ( or for the complex case ).

A fundamental part of ` Abdul-Bahá's teachings on evolution is the belief that all life came from the same origin: " the origin of all material life is one ..." He states that from this sole origin, the complete diversity of life was generated: " Consider the world of created beings, how varied and diverse they are in species, yet with one sole origin " He explains that a slow, gradual process led to the development of complex entities:

* Consider the C *- algebra of complex square matrices.

Consider for example any compact connected complex manifold M: any holomorphic function on it is locally constant by Liouville's theorem.

Consider a complex scalar field φ, with the constraint that φ < sup >*</ sup > φ = v², a constant.

Consider a complex, real-world problem, like those of marketing or making policies for a nation, where there are many governing factors, and most of them cannot be expressed as numerical time series data, as one would like to have for building mathematical models.

Consider a once-punctured elliptic curve, given as the locus D of complex points satisfying, where and is a complex number.

Consider a d < sup > 6 </ sup > octahedral complex ( example IrBr < sub > 6 </ sub >< sup > 3 -</ sup >).

* Consider C, the field of complex numbers, as a 1-dimensional vector space.

Consider a polygon in the complex plane.

Consider the logarithm function: For any fixed base b, the logarithm function log < sub > b </ sub > maps from the positive real numbers R < sup >+</ sup > onto the real numbers R ; formally:

Consider a wave packet as a function of position x and time t: α ( x, t ).

Consider a pseudo random number generator ( PRNG ) function P ( key ) that is uniform on the interval 2 < sup > b </ sup > − 1.

Consider a function from a metric space M to a topological space V, and a point c of M. We direct the set M

Consider a function that takes no parameters and returns input from the keyboard.

Consider a wave function that is a sum of many waves, however, we may write this as

Consider a function with its corresponding graph as a subset of the Cartesian product.

Consider the function f, piecewise defined by f ( x ) = – 1 for x < 0 and f ( x ) = 1 for x ≥ 0.

Consider the recursion equations for the factorial function f:

Consider a function that reads the next line of text from a given file:

Consider a function of jump process.

Consider a graph G with vertices V, each numbered 1 through N. Further consider a function shortestPath ( i, j, k ) that returns the shortest possible path from i to j using vertices only from the set

Consider a differentiable function ƒ ( x ) whose derivative is ƒ '( x ).

Consider the vector-valued function F from R < sup > 2 </ sup > to R < sup > 2 </ sup > defined by

Consider an open set on the real line and a function f defined on that set with real values.

Consider, for purposes of illustration, a mountainous landscape M. If f is the function sending each point to its elevation, then the inverse image of a point in ( a level set ) is simply a contour line.

Consider two waveforms f and g. By calculating the convolution, we determine how much a reversed function g must be shifted along the x-axis to become identical to function f. The convolution function essentially reverses and slides function g along the axis, and calculates the integral of their ( f and the reversed and shifted g ) product for each possible amount of sliding.

Consider the function

The Mind in Consider Phlebas is also described as having internal power sources which function as back-up shield generators and space propulsion, and seeing the rational, safety-conscious thinking of Minds, it would be reasonable to assume that all Minds have such features, as well as a complement of drones and other remote sensors as also described.

Consider a 2 × 2 banyan switch, which requires ( 2 / 2 ) log < sub > 2 </ sub > 2

#: Consider a unit sphere placed at the origin, a rotation around the x, y or z axis will map the sphere onto itself, indeed any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis, see Euler angles.

Consider a point charge q with position ( x, y, z ).

Consider a Lorentz boost in a fixed direction z.

Consider a light wave propagating along the z principal axis polarised such the electric field of the wave is parallel to the x-axis.

Consider the projection π: S < sup > 1 </ sup > → S < sup > 1 </ sup > given by z ↦ z < sup > 2 </ sup >.

Consider the Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riemann sphere with values in C, which by Liouville's theorem is constant.

Consider the Laurent series of f ( z ) about i, the only singularity we need to consider.