Mathematical Methods for Physicists: A concise introduction - Site Map

ELEMENTARY FUNCTIONS OF z
Setting w ˆ u ‡ iv and z ˆ re i ˆjzje i we have
It follows that
and
Therefore
e w ˆ e u‡iv ˆ e u e iv ˆ re i :
e u ˆ r ˆ jj z or u ˆ ln r ˆ ln jj z
v ˆ ˆ arg z:
w ˆ ln z ˆ ln r ‡ i ˆ ln jj‡i z arg z:
Since the argument of z is determined only in multiples of 2, the complex
natural logarithm is in®nitely many-valued. If we let 1 be the principal argument
of z, that is, the particular argument of z which lies in the interval 0 0. Now we can take a
natural logarithm of a negative number, as shown in the following example.
Example 6.10
ln 4 ˆ lnj4j‡i arg…4† ˆln 4 ‡ i… ‡ 2n†; its principal value is ln 4 ‡ i…†,
a complex number. This explains why the logarithm of a negative number makes
no sense in real variable.
Hyperbolic functions
We conclude this section on ``elementary functions'' by mentioning brie¯y the
hyperbolic functions; they are de®ned at points where the denominator does not
vanish:
sinh z ˆ 1
2 …ez e z †; cosh z ˆ 1
2 …ez ‡ e z †;
tanh z ˆ sinh z=cosh z; coth z ˆ cosh z=sinh z;
sech z ˆ 1=cosh z; cosech z ˆ 1=sinh z:
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