Figure 1 shows the graph of .
Instead of many mathematicians write , physicists (and calculators) however consider as the symbol for the logarithm in base 10. One can show that the function defined in this way is the inverse of the exponential function. Indeed with equation (1) we have
so there exists such that
Now applying the substitution law with we have
so we have
The natural logarithm can also be represented as a power series around . For we have
In real analysis there is no reasonable way to extend the logarithm to negative numbers. In complex analysis the situation is a bit more complicated. Basically one can use the Euler relation to write a non-zero complex number in the form with . We could try to define the complex logarithm to be . However is unique only up to addition of a multiple of . While at first glance this does not appear to be very problematic, it actually prevents one from setting up a continuous logarithm on the complex plane (without 0, where the logarithm should be infinite). Say for example that we let the imaginary part of our logarithm take values from to . Then
while for both limits. Therefore the logarithm we defined is not continuous at -1. The same argument allows one to show that it is not continuous on the negative real numbers. In fact you can only define a continuous complex logarithm on a sliced plane, i.e. the complex plane with a half-line starting at 0 removed.
|Date of creation||2013-03-22 12:28:28|
|Last modified on||2013-03-22 12:28:28|
|Last modified by||mathwizard (128)|