Complex Numbers Exponential - Real Exponents

Real Exponents

The identities and properties shown above for integer exponents are true for positive real numbers with non-integer exponents as well. However the identity

cannot be extended consistently to where b is a negative real number, see Real exponents with negative bases. The failure of this identity is the basis for the problems with complex number powers detailed under failure of power and logarithm identities.

The extension of exponentiation to real powers of positive real numbers can be done either by extending the rational powers to reals by continuity, or more usually by using the exponential function and its inverse, the natural logarithm.

Read more about this topic:  Complex Numbers Exponential

Famous quotes containing the word real:

    A decent chap, a real good sort,
    Straight as a die, one of the best,
    A brick, a trump, a proper sport,
    Head and shoulders above the rest;
    How many lives would have been duller
    Had he not been here below?
    Here’s to the whitest man I know
    Though white is not my favourite colour.
    Philip Larkin (1922–1986)