The document discusses using imaginary numbers and derivatives to evaluate the Birch and Swinnerton-Dyer conjecture, which relates elliptic curves to special values of L-functions. It shows applying the product rule formula to take the derivative of a function F(r) involving imaginary numbers. Through integration by parts, it arrives at the solution L(C,s) = c(-2) + C, relating the L-function to constants.

1. Imaginary Numbers of the Birch and Swinnerton-
Dyer Conjecture
L(C,s)= c(s-1)r
s=1, r=2, -1=(i)2
, & c≠0
L(C,s) = c(1+(i)2
)2
L(C,s)=c(12
+2(i)2
+i4
)
r is inserted for 2, hence,
F(r) = c(1r
+r(i)r
+1)
d (1r
)/dr= (1r
)(ln1), ln1=0, therefore, d (1r
)/dr = 0,

2. & d (1)/dr = 0, then one calculates the derivative of
r(i)r
via the Product Rule.
Product Rule
F´(r)= c (f(r)g´(r) + f´(r)g(r))
f(r)=r, g´(r)= (i)r
lni, f´(r)=1, g(r)= (i)r
F´(r)= c ((r)(i)r
lni+(i)r
)
∫ F´(r) dr= c(∫(r)(i)r
lni dr + ∫(i)r
dr)
= c (∫f(r)g´(r)dr + ∫(i)r
dr) +C

3. Integration by Parts
∫f(r)g´(r)dr= c(f(r)g(r)- ∫f´(r)g(r)dr)
=c(f(r)g(r) - ∫f´(r)g(r)dr + ∫(i)r
dr)
f(r)=r, f´(r)=1, g(r)=(i)r
, g´(r)=(i)r
lni
=c(r(i)r
– ∫(i)r
dr + ∫(i)r
dr) + C
=c[(2)(-1)] + C
L(C,s) = c(-2) + C