The formula may also be obtained by repeated integration by parts, and the leading term can be found through Laplace's method.
12.
In terms of the roots-and-leading-term formula, multiplying all the coefficients by does not change the roots, but multiplies the leading term by.
13.
But by hypothesis this reduction for leaves its leading term, of the form for a non-zero constant, as the only nonzero term.
14.
P _ c ^ n ( 0 ) is a polynomial in c and its leading terms settle down as n grows large enough.
15.
But any multiple in of has leading term divisible by " q " and constant term divisible by " p ", which proves the statement.
16.
Dividing both parts by and taking the limit will produce, the coefficient of the highest-order term in the expansion, which represents the rate at which changes in its leading term.
17.
The leading term in this power series is the solution of the exactly solvable problem, while further terms describe the deviation in the solution, due to the deviation from the initial problem.
18.
The leading term of D ( x ) is obtained by shifting the contour past the double pole at w = 1 : the leading term is just the residue, by Cauchy's integral formula.
19.
The leading term of D ( x ) is obtained by shifting the contour past the double pole at w = 1 : the leading term is just the residue, by Cauchy's integral formula.
20.
But then necessarily the reductions modulo of and also make all non-leading terms vanish ( and cannot make their leading terms vanish ), since no other decompositions of are possible in, which is a unique factorization domain.
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