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12.6 Trigonometric Functions
Next we calculate
for
.
Now
and it is clear that
,
is pure imaginary. Hence,
i.e.,
![\begin{displaymath}
\exp(it)=\cos t+i\sin t\mbox{ for all } t\in\mbox{{\bf R}}.
\end{displaymath}](img1904.gif) |
(12.46) |
For any complex number
, we have
Since your calculator has buttons that calculate approximations to
,
and
, you can approximately calculate the exponential of any complex number
with a few key strokes.
The relation (12.46)
actually holds for all
, since
Hence
![\begin{displaymath}
e^{iz}=\cos z+i\sin z\mbox{ for all }z\in\mbox{{\bf C}},
\end{displaymath}](img1909.gif) |
(12.47) |
so
![\begin{displaymath}
e^{-iz}=\cos z-i\sin z\mbox{ for all }z\in\mbox{{\bf C}}.
\end{displaymath}](img1910.gif) |
(12.48) |
We can solve (12.47) and (12.48) for
and
to obtain
![\begin{displaymath}
\cos(z)={{e^{iz}+e^{-iz}}\over 2}\mbox{ for all }z\in\mbox{{\bf C}}.
\end{displaymath}](img1913.gif) |
(12.49) |
![\begin{displaymath}
\sin(z)={{e^{iz}-e^{-iz}}\over {2i}}\mbox{ for all }z\in\mbox{{\bf C}}.
\end{displaymath}](img1914.gif) |
(12.50) |
From (12.47) it follows that
i.e.,
is in the unit circle for all
.
12.51
Exercise (Addition laws for
and
.)
A
Prove that
for all
![$z,a\in\mbox{{\bf C}}$](img1918.gif)
.
By the addition laws, we have (for all
),
![\begin{displaymath}
\cos(x+iy)=\cos (x)\cos(iy)-\sin (x)\sin(iy)
\end{displaymath}](img1920.gif) |
(12.52) |
![\begin{displaymath}
\sin(x+iy)=\sin x\cos(iy)+\cos x\sin(iy).
\end{displaymath}](img1921.gif) |
(12.53) |
By (12.49) and (12.50)
and
12.54
Definition (Hyperbolic functions.)
For all
![$z\in\mbox{{\bf C}}$](img66.gif)
, we define the
hyperbolic sine and
hyperbolic cosine of
![$z$](img795.gif)
by
Note that if
is real,
and
are real. Most calculators have
buttons that calculate
and
. We can now rewrite (12.52) and
(12.53) as
These formulas hold true for all complex
and
.
Since
it follows from our
discussion in example 10.45 that
for all
. In particular
and
Hence
on
, so
is strictly decreasing on
.
Moreover
is continuous (since it is differentiable) so by the intermediate
value theorem there is a number
in
such that
. Since
is strictly decreasing on
this number
is unique. (Cf. exercise
5.48.)
12.55
Definition (
.)
We define the real number
![$\pi$](img1940.gif)
by the condition
![$\displaystyle {{\pi\over 2}}$](img1941.gif)
is the unique
number in
![$(0,2)$](img1937.gif)
satisfying
![$\displaystyle {\cos\left({\pi\over 2}\right)=0}$](img1942.gif)
.
12.56
Theorem.
is periodic of period
; i.e.,
Proof: Since
for all
, we have
, so
. We have noted that
on
so
. Hence
and
![\begin{displaymath}
e^{2i\pi}=\left(e^{{{i\pi}\over 2}}\right)^4 =i^4=1.
\end{displaymath}](img1951.gif) |
(12.57) |
It follows that
for all
.
12.58
Entertainment.
If Maple or Mathematica are asked for the numerical values of
![$(-1)^{3.14}$](img1953.gif)
and
![$i^i$](img1954.gif)
, they agree that
and
Can you propose a reasonable definition for
![$(-1)^z$](img1957.gif)
and
![$i^z$](img1958.gif)
when
![$z$](img795.gif)
is an
arbitrary complex number, that is consistent with these results? To be reasonable
you would require that when
![$z\in\mbox{{\bf Z}}$](img1959.gif)
,
![$(-1)^z$](img1957.gif)
and
![$i^z$](img1958.gif)
give the expected values,
and
Proof: From the previous exercise,
. We've noted that
for
,
Hence
for
. Hence
for
. Hence
is strictly decreasing on
. Hence
for all
.
Now
and since
, we've shown that
for all
.
12.61
Theorem.
Every point
in the unit
circle can be written as
for a unique
.
Proof: We first show uniqueness.
Suppose
where
. Without loss of
generality, say
. Then
and
. By the previous theorem,
is the only number in
whose cosine is
, so
, and hence
.
Let
be a point in the unit circle, so
, and hence
. Since
and
, it follows from the intermediate value
theorem that
for some
. Then
so
.
and since
, we have
.
12.62
Lemma.
The set of all complex solutions
to
is
.
Proof: By exercise 12.59A
so
Let
be any solution to
; i.e.,
By uniqueness of polar decomposition,
so
(since for real
,
). We can write
where
and
by theorem
5.30 ,
so
where
. Now
By theorem 12.61,
, so
, and
;
i.e.,
.
12.63
Definition (Argument.)
Let
![$a\in\mbox{{\bf C}}\backslash\{0\}$](img1033.gif)
and write
![$a$](img590.gif)
in its polar decomposition
![$a=\vert a\vert u$](img2025.gif)
,
where
![$\vert u\vert=1$](img2026.gif)
. We know
![$u=e^{iA}$](img2027.gif)
for a unique
![$A\in[0,2\pi)$](img2028.gif)
. I will call
![$A$](img407.gif)
the
argument of
![$a$](img590.gif)
and write
![$A=\mbox{{\rm Arg}}(a)$](img2029.gif)
. Hence
12.64
Remark.
Our definition of
![$\mbox{{\rm Arg}}$](img2032.gif)
is rather arbitrary. Other natural definitions are
or
None of these argument functions is continuous; e.g.,
But
Proof: Since
the numbers given are solutions to
. Let
be any solution to
.
Then
. Hence, by the lemma 12.62,
We will now look at
geometrically as a function from
to
.
Claim:
maps the vertical line
into the circle
.
Proof: If
, then
Claim:
maps the horizontal line
into the ray through
with
direction
.
Proof: If
, then
Since
is periodic of period
,
maps an infinite horizontal
strip of width
into an infinite circular segment making `` angle
" at the
origin.
The Exponentials of Some Cats
maps every strip
onto all of
.
Proof: These numbers are clearly solutions to
. Let
be
any solution to
. Then
By uniqueness of polar decomposition,
i.e.,
and
. Hence,
for some
and
Thus
For each
, the number
is a solution to
. For
, the numbers
are distinct numbers in
, so the numbers
are
distinct. For every
,
where
and
, so
where
and
; i.e.,
Then
, so
Next: 12.7 Special Values of
Up: 12. Power Series
Previous: 12.5 Logarithms
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