A canonical form for antiderivatives

A Canonical Form for Circuit Diagrams

A canonical form for circuit diagrams with a designated start state is presented. The form is based upon nite automata minimization and Shannon's canonical form for boolean expressions.

Antiderivatives for Complex Functions

Since we have the same product rule, quotient rule, sum rule, chain rule etc. available to us for differentiating complex functions, we already know many antiderivatives. For example, by differentiating f (z) = z one obtains f ′(z) = nzn−1, and from this one sees that the antiderivative of z is 1 n+1 z – except for the very important case where n = −1. Of course that special case is very import...

A Rational Canonical Form Algorithm

Jordan Canonical Form

is the geometric multiplicity of λk which is also the number of Jordan blocks corresponding to λk . • The orders of the Jordan Blocks of λk must sum to the algebraic multiplicity of λk . • The number of Jordan blocks corresponding to an eigenvalue λk is its geometric multiplicity. • The matrix A is diagonalizable if and only if, for any eigenvalue λ of A , its geometric and algebraic multiplici...

The Jordan Canonical Form

Let β1, . . . , βn be linearly independent vectors in a vector space. For all j with 0 ≤ j ≤ n and all vectors α1, . . . , αk, if β1, . . . , βn are in the span of β1, . . . , βj, α1, . . . , αk, then j + k ≥ n. The proof of the claim is by induction on k. For k = 0, the claim is obvious since β1, . . . , βn are linearly independent. Suppose the claim is true for k−1, and suppose that β1, . . ....