• I. N. Bronshtein
  • K. A. Semendyayev


Definitions. An algebraic expression is a collection of one or more algebraic quantities (numbers or letters) connected by the signs of operations (+, -, :, √ etc.) with brackets of different kinds indicating the succession of operations.


Algebraic Equation Canonical Form Real Root Transcendental Equation Multiple Root 


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  1. (1).
    For equations see p. 158.Google Scholar
  2. (1).
    See p. 164.Google Scholar
  3. (1).
    I.e., the term containing x in the highest power.Google Scholar
  4. (1).
    I.e., denominator and numerator have no common factor containing x.Google Scholar
  5. (2).
    If we do not confine ourselves to the real numbers, then the case (3) will not differ from (1) and the case (4) from (2). From this point of view, each fraction R(x) can be transformed into a sum of partial fractions of the form \(\frac{A}{(x-\alpha )^{k}}\), where A and α are complex numbers. This is applied in solving linear differential equations (see p. 543).Google Scholar
  6. (3).
    For simple and multiple roots see p. 164.Google Scholar
  7. (1).
    The following formula is easy to remember \(\textup{log}_{a}\;N=\frac{\textup{log}\;N}{\textup{log}\;a}\), where the logarithms on the right hand side are taken to an arbitrary, but to the same base.Google Scholar
  8. (2).
    For the number e see p. 331.Google Scholar
  9. (1).
    For tables of common logarithms see pp. 50–52; tables of antilogarithms pp. 53–55; tables of natural logarithms pp. 68–71.Google Scholar
  10. (2).
    To find a logarithm of an expression involving addition and subtraction, this expression should be first transformed into a suitable form (i.e., into a form containing multiplication and division).Google Scholar
  11. (1).
    An equality that is true for arbitrary values of the variable x is called an identity.Google Scholar
  12. (2).
    Here and in the sequel, the coefficients a0, a1,…, an are assumed to be real, with exception of certain cases which are discussed separately.Google Scholar
  13. (1).
    For multiple roots see p. 164.Google Scholar
  14. (1).
    The coefficient a0 at the highest power of x has been made equal to 1 (by dividing the equation by this coefficient).Google Scholar
  15. (1).
    For conjugate complex numbers see p. 587.Google Scholar
  16. (2).
    A way of obtaining such an equation has been given above. Practically, one can leave the multiple roots and at once form Sturm sequence of functions; if the last remainder Pm is not a constant, then P(x) has multiple roots and they should be abandoned.Google Scholar
  17. (1).
    To simplify the computation, the obtained remainders can be multiplied by constant positive factors; this will not change the result.Google Scholar
  18. (2).
    If some of these number are zero, then they should be omitted in counting the changes of sign.Google Scholar
  19. (1).
    For continuity of a function see p. 335.Google Scholar
  20. (1).
    The first index, i, of the element aij indicates that the element is taken from the i-th row of the determinant; the second index, j, indicates that the element is taken from the j-th column (i—number of the row counted from the top; j—number of the column counted from the left).Google Scholar
  21. (2).
    For permutations see p. 192.Google Scholar
  22. (1).
    A system of solutions of the form x1 = α1, x2 = α2, …, xn = αn will be denoted by α1, α2, …, αn.Google Scholar
  23. (1).
    See footnote on p. 177.Google Scholar
  24. (2).
    There can be infinitely many basic systems of solutions (see below).Google Scholar
  25. (1).
    Degree of a polynomial of two variables x and y is the highest sum of exponents of these variables in members of the polynomial. For example, the polynomial x3 + x2y2 + y3 is of the fourth degree.Google Scholar
  26. (1).
    If the symbol (3) concerns the values for which the relations of “greater” or “less” are not defined, e.g., complex numbers (p. 585), vectors (p. 613), then it can not be replaced by the symbol (3a). In this section, we are concerned only with real numbers.Google Scholar
  27. (1).
    For a particular case (n = 2) of the inequality see p. 190.Google Scholar
  28. (1).
    If n = 3, then a1, a2, a3 and b1, b2, b3 may be regarded as rectangular Cartesian coordinates of a vector; then Buniakowsky-Cauchy inequality reads that the scalar product of vectors does not exceed the product of their moduli (see p. 617). For n > 3, this formulation is extended to vectors in the n-dimensional space. An analogy of Buniakowsky-Cauchy inequality for infinite convergent series: \(\left ( \sum_{i-1}^{\infty} a_{i}b_{i}\right )^2\leqslant \left ( \sum_{i-1}^{\infty} a_{i}^{2} \right )\left ( \sum_{i-1}^{\infty} b_{i}^{2} \right )\) An analogy of the same inequality for definite integrals: \(\left (\int\limits_{a}^{b}f(x)\varphi(x)dx \right)^2\leqslant \left (\int\limits_{a}^{b}[f(x)]^2dx \right)\left (\int\limits_{a}^{b}[\varphi (x)]^2dx \right)\).Google Scholar
  29. (1).
    For a table of infinite series see pp. 353–354.Google Scholar
  30. (1).
    And also to the complex numbers.Google Scholar
  31. (2).
    For complex x, if re (x) >0.Google Scholar
  32. (1).
    For the symbol “n!” (the factorial) see p. 190.Google Scholar

Copyright information

© Verlag Harri Deutsch, Zürich 1973

Authors and Affiliations

  • I. N. Bronshtein
  • K. A. Semendyayev

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