TABLE OF CONTENTS This appendix gives various mathematical relations that are used regularly in the calculations of the text. The references listed at the end of this appendix contain extensive tables of other useful mathematical relations and analytic expressions.
The quadratic equation
has the two solutions
The cube equation
can be reduced to the form
by substituting for z the value
The constants a and b are given by the expressions
Let A and B be defined as
The three roots of (A.62) are given by the following expressions
Let
If ? > 0, then there will be one real root and two complex conjugate roots.
If ? = 0, there will be three real roots, of which at least two are equal. If ? < 0, there will be three real and unequal roots.
Let f( x, t) be continuous and have a continuous derivative ? f/ ? t, in a domain in the x-t plane that includes the rectangle
In addition, let ?( t) and ?( t) be defined and have continuous first derivatives for t 1 ? t ? t 2
