Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
functions cos z and sec z are even, and the other functions are odd. Since the
exponential function is periodic, the trigonometric functions are also periodic,
and we have
cos…z 2n† ˆcos z; sin…z 2n† ˆsin z;
tan…z 2n† ˆtan z; cot…z 2n† ˆcot z;
where n ˆ 0; 1; ...:
Another important property also carries over: sin z and cos z have the same
zeros as the corresponding real-valued functions:
sin z ˆ 0 if and only if z ˆ n …n integer†;
cos z ˆ 0 if and only if z ˆ…2n ‡ 1†=2 …n integer†:
We can also write these functions in the form u…x; y†‡iv…x; y†. As an example,
we give the details for cos z. From Eq. (6.19) we have
cos z ˆ 1
2 …eiz ‡ e iz †ˆ1
2 …ei…x‡iy† ‡ e i…x‡iy† †ˆ1
2 …ey e ix ‡ e y e ix †
ˆ 1
2 ‰ey …cos x ‡ i sin x†‡e y …cos x i sin x†Š
ˆ cos x ey ‡ e y
i sin x ey e y
2
2
or, using the de®nitions of the hyperbolic functions of real variables
cos z ˆ cos…x ‡ iy† ˆcos x cosh y i sin x sinh y;
similarly,
sin z ˆ sin…x ‡ iy† ˆsin x cosh y ‡ i cos x sinh y:
In particular, taking x ˆ 0 in these last two formulas, we ®nd
cos…iy† ˆcosh y; sin…iy† ˆi sinh y:
There is a big di€erence between the complex and real sine and cosine functions.
The real functions are bounded between 1 and ‡1, but the
complex functions can take on arbitrarily large values. For example, if y is real,
then cos iy ˆ 1
2 …ey ‡ e y †!1as y !1or y !1.
The logarithmic function w ˆ ln z
The real natural logarithm y ˆ ln x is de®ned as the inverse of the exponential
function e y ˆ x. For the complex logarithm, we take the same approach and
de®ne w ˆ ln z which is taken to mean that
e w ˆ z
…6:20†
for each z 6ˆ 0.
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