Orders of Magnitude: The 0, o, ~ Notation

  • Richard M. Meyer
Part of the Universitext book series (UTX)


In many practical applications of Mathematics it is necessary to consider the behavior of some function f(x) of x as x tends to some limit (finite or infinite). However, many times the function f(x) is extremely complicated in nature, or incompletely known, and it is preferable (in fact, perhaps only possible) to describe the asymptotic behavior of f(x) relative to (or compared with) some other function g(x) of x as x tends to the same limit. In practice, the comparison function g is often chosen as a “simpler” function, such as a power or exponential function.


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References to Additional and Related Material: Section 6

  1. 1.
    Abramowitz, M. and I. Stegun (Editors), “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables”, National Bureau of Standards, Applied Mathematics Series No. 55, Washington, D. C. (1964).Google Scholar
  2. 2.
    Brand, L.,“Advanced Calculus”, Wiley and Sons, Inc., New York (1958).Google Scholar
  3. 3.
    Courant, R., “Differential and Integral Calculus” Vol. I, (Second Ed.), Blackie and Son, Ltd., London (1963).Google Scholar
  4. 4.
    Cramér, H., “Mathematical Methods of Statistics”, Princeton University Press (1958).Google Scholar
  5. 5.
    Hobson, E., “Theory of Functions of a Real Variable”, Vol. II, Dover Publications, New York.Google Scholar
  6. 6.
    Lebedev, N., “Special Functions and Their Applications”, Prentice-Hall, Inc. Englewood Cliffs, N.J. (1965).zbMATHGoogle Scholar
  7. 7.
    Pearson, K., (Editor), “Tables of the Incomplete Gamma Function”, London (1922).Google Scholar
  8. 8.
    Richardson, C., “An Introduction to the Calculus of Finite Differences”, Van Nostrand, Inc., New York (1954).zbMATHGoogle Scholar
  9. 9.
    Spiegel, M., “Mathematical Handbook of Formulas and Tables”, Schaum’s Outline Series, Schaum Publishing Co., New York (1968).Google Scholar
  10. 10.
    Whittaker, E. and G. Watson, “A Course in Modern Analysis”, Cambridge University Press (1952).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1979

Authors and Affiliations

  • Richard M. Meyer
    • 1
  1. 1.Niagara University, College of Arts and SciencesNiagara UniversityUSA

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