Sometimes you may find the notation F ( ϕ , k ) in (12.2) used when k > 1. Allowing this notation, show that 1 3 F sin − 1 3 5 , 4 7 = 1 4 F sin − 1 4 5 , 3 4 . Hints: Using the Jacobi form of F in ( 12.2 ) , write the integral which is equal to 1 3 F sin − 1 3 5 , 4 3 . Follow Example 3 to make a change of variable, write the corresponding integral, and verify that it is equal to 1 4 F sin − 1 4 5 , 3 4 .
Sometimes you may find the notation F ( ϕ , k ) in (12.2) used when k > 1. Allowing this notation, show that 1 3 F sin − 1 3 5 , 4 7 = 1 4 F sin − 1 4 5 , 3 4 . Hints: Using the Jacobi form of F in ( 12.2 ) , write the integral which is equal to 1 3 F sin − 1 3 5 , 4 3 . Follow Example 3 to make a change of variable, write the corresponding integral, and verify that it is equal to 1 4 F sin − 1 4 5 , 3 4 .
Sometimes you may find the notation
F
(
ϕ
,
k
)
in (12.2) used when
k
>
1.
Allowing this notation, show that
1
3
F
sin
−
1
3
5
,
4
7
=
1
4
F
sin
−
1
4
5
,
3
4
.
Hints: Using the Jacobi form of
F
in
(
12.2
)
,
write the integral which is equal to
1
3
F
sin
−
1
3
5
,
4
3
.
Follow Example 3 to make a change of variable, write the corresponding integral, and verify that it is equal to
1
4
F
sin
−
1
4
5
,
3
4
.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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