Verify Bauer’s formula e i x w = ∑ 0 ∞ ( 2 l + 1 ) i i j i ( x ) P l ( w ) as follows. Write the integral for the coefficients a in the Legendre series for e i x w = ∑ c i P i ( w ) . You want to show that c l ( x ) = ( 2 l + 1 ) i l j l ( x ) . First show that y = c l ( x ) satisfies the differential equation (Problem 17.6) for spherical Bessel functions. Hints: Differentiate withrespect to x under the integral sign to find y ′ =of the differential equation. Now integrate by parts with respect to w to show =hat the integrand is zero because P l ( w ) satisfies Legendre’s equation. Thus c l ( x ) enust be a linear combination of j ( x ) and n 1 ( x ) . Now consider the c 1 ( x ) integral for small x; expand e i w x in series and evaluate the lowest term (which is x l since ∫ − 1 1 w n P l ( w ) d w = 0 for n < l ). Compare with the approximate formulas for j 1 ( x ) and n l ( x ) in Section 20.
Verify Bauer’s formula e i x w = ∑ 0 ∞ ( 2 l + 1 ) i i j i ( x ) P l ( w ) as follows. Write the integral for the coefficients a in the Legendre series for e i x w = ∑ c i P i ( w ) . You want to show that c l ( x ) = ( 2 l + 1 ) i l j l ( x ) . First show that y = c l ( x ) satisfies the differential equation (Problem 17.6) for spherical Bessel functions. Hints: Differentiate withrespect to x under the integral sign to find y ′ =of the differential equation. Now integrate by parts with respect to w to show =hat the integrand is zero because P l ( w ) satisfies Legendre’s equation. Thus c l ( x ) enust be a linear combination of j ( x ) and n 1 ( x ) . Now consider the c 1 ( x ) integral for small x; expand e i w x in series and evaluate the lowest term (which is x l since ∫ − 1 1 w n P l ( w ) d w = 0 for n < l ). Compare with the approximate formulas for j 1 ( x ) and n l ( x ) in Section 20.
Verify Bauer’s formula
e
i
x
w
=
∑
0
∞
(
2
l
+
1
)
i
i
j
i
(
x
)
P
l
(
w
)
as follows. Write the integral for the coefficients
a
in the Legendre series for
e
i
x
w
=
∑
c
i
P
i
(
w
)
.
You want to show that
c
l
(
x
)
=
(
2
l
+
1
)
i
l
j
l
(
x
)
.
First show that
y
=
c
l
(
x
)
satisfies the differential equation (Problem 17.6) for spherical Bessel functions. Hints: Differentiate withrespect to
x
under the integral sign to find
y
′
=of the differential equation. Now integrate by parts with respect to
w
to show =hat the integrand is zero because
P
l
(
w
)
satisfies Legendre’s equation. Thus
c
l
(
x
)
enust be a linear combination of
j
(
x
)
and
n
1
(
x
)
.
Now consider the
c
1
(
x
)
integral for small x; expand
e
i
w
x
in series and evaluate the lowest term (which is
x
l
since
∫
−
1
1
w
n
P
l
(
w
)
d
w
=
0
for
n
<
l
). Compare with the approximate formulas for
j
1
(
x
)
and
n
l
(
x
)
in Section 20.
3.
X
1 x)² dx [₁1lux ?
5x₁₂ (2+inx) +C
(2 + ln x)³
A. (2 + ln x) 4 +C
C. (2 + In x)³ + C
B. (2 + Inx) + C
D. (2 + In x)² + C
= 0.70
using
five iterations of the Bisection Method on
ln(a*) :
1. Find one real root of
(0.5, 2).
the interval
2. Determine one real root of
2xcos2x – (x – 2)² = 0
on the
-
interval
(2,3)using the Regula-Falsi
Method. Do four iterations.
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