In the generating function equation of Problem 19, put h = e i θ and separate real and imaginary parts to derive the equations cos ( x sin θ ) = J 0 ( x ) + 2 J 2 ( x ) cos 2 θ + 2 J 4 ( x ) cos 4 θ + ⋯ = J 0 ( x ) + 2 ∑ n = 1 J 2 n ( x ) cos 2 n θ , sin ( x sin θ ) = 2 J 1 ( x ) sin θ + J 3 ( x ) sin 3 θ + ⋯ = 2 ∑ n = 0 ∞ J 2 n + 1 ( x ) sin ( 2 n + 1 ) θ . These are Fourier series with Bessel functions as coefficients. (In fact the J n ’s for integral n are often called Bessel coefficients because they occur in many series like these.) Use the formulas for the coefficients in a Fourier series to find integrals representing J n for even n and for odd n . Show that these results can be combined to give J n ( x ) = 1 π ∫ 0 π cos ( n θ − x sin θ ) d θ for all integral n. These series and integrals are of interest in astronomy and in the theory of frequency modulate waves.
In the generating function equation of Problem 19, put h = e i θ and separate real and imaginary parts to derive the equations cos ( x sin θ ) = J 0 ( x ) + 2 J 2 ( x ) cos 2 θ + 2 J 4 ( x ) cos 4 θ + ⋯ = J 0 ( x ) + 2 ∑ n = 1 J 2 n ( x ) cos 2 n θ , sin ( x sin θ ) = 2 J 1 ( x ) sin θ + J 3 ( x ) sin 3 θ + ⋯ = 2 ∑ n = 0 ∞ J 2 n + 1 ( x ) sin ( 2 n + 1 ) θ . These are Fourier series with Bessel functions as coefficients. (In fact the J n ’s for integral n are often called Bessel coefficients because they occur in many series like these.) Use the formulas for the coefficients in a Fourier series to find integrals representing J n for even n and for odd n . Show that these results can be combined to give J n ( x ) = 1 π ∫ 0 π cos ( n θ − x sin θ ) d θ for all integral n. These series and integrals are of interest in astronomy and in the theory of frequency modulate waves.
In the generating function equation of Problem 19, put
h
=
e
i
θ
and separate real and imaginary parts to derive the equations
cos
(
x
sin
θ
)
=
J
0
(
x
)
+
2
J
2
(
x
)
cos
2
θ
+
2
J
4
(
x
)
cos
4
θ
+
⋯
=
J
0
(
x
)
+
2
∑
n
=
1
J
2
n
(
x
)
cos
2
n
θ
,
sin
(
x
sin
θ
)
=
2
J
1
(
x
)
sin
θ
+
J
3
(
x
)
sin
3
θ
+
⋯
=
2
∑
n
=
0
∞
J
2
n
+
1
(
x
)
sin
(
2
n
+
1
)
θ
.
These are Fourier series with Bessel functions as coefficients. (In fact the
J
n
’s for integral
n
are often called Bessel coefficients because they occur in many series like these.) Use the formulas for the coefficients in a Fourier series to find integrals representing
J
n
for even
n
and for odd n. Show that these results can be combined to give
J
n
(
x
)
=
1
π
∫
0
π
cos
(
n
θ
−
x
sin
θ
)
d
θ
for all integral n. These series and integrals are of interest in astronomy and in the theory of frequency modulate waves.
3. An alternating current-direct current (AC-DC) voltage signal is made up of the following
two components, each measured in volts (V): VAC = 10 sint, VDc = 15. [4]
a) What types of functions are VAC and VDC?
b) Determine the equation of the combined function VAC + VDC.
c) Determine the domain and range of the combined function.
Find the range of y =cos4x - 1.
1
sys
2
1
2
-5 sys 3
3
3
sys
2
1
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