Sentence examples for the outer integral from inspiring English sources

The double integral can then be rewritten to express the inner integral in terms of v and the outer integral in terms of s′.

for z ∈ D Open image in new window, where z 0 = c - 1 ( 0 ) Open image in new window, the outer integral is over any curve connecting z0and z, and c = u0 z1) - u s (z1).

for z ∈ D Open image in new window, where z 0 = c - 1 ( 0 ) Open image in new window, the outer integral is over any curve connecting z0 and z, and c = u0 z1) - u s (z1), z1 being an arbitrary point on the smooth part of S Open image in new window.

for z ∈ D a, φ Open image in new window, where z 0 = c a, φ - 1 ( 0 ) Open image in new window, the outer integral is over any path in D a, φ Open image in new windowconnecting z0and z, and c = u0 z1) - u s (z1), z1being an arbitrary point on the smooth part of S Open image in new window.

for z ∈ D a, φ Open image in new window, where z 0 ∈ D a, φ Open image in new windowis fixed and the outer integral is over any path in D a, φ Open image in new windowconnecting z0and z.

also holds for arbitrary function f ∈ L p (v), which follows from (1.6) considered for the function g ( t ) = f ( t ) χ ( a, a + ε ) ( t ) and then from the monotonicity of the outer integral on the left hand side of (1.6).

ℋ, one immediately has ∫ | t | ≥ n 1 / 4 2 M | d l d t l φ n ( t ) | 2 d t ≤ E [ ∫ | t | ≥ n 1 / 4 2 M | d l d t l ∏ j = 1 n E [ e i t f ˜ ( X j ) / n ∣ H ] | 2 d t ] ≤ β n, l. for every n ∈ N and l = 0, 1, which concludes the study of the outer integrals by means of (12).

This technique provides an efficient method for the computation of the inner angular integral; however, the outer semi-infinite integral still converges very slowly when a real axis integration is applied.

In this paper, it is shown that this problem can be overcome by using the ILHI representation for the angular integral to develop a novel asymptotic extraction technique for the outer semi-infinite integral.

The incomplete Lipschitz-Hankel integral representation for the angular integral is then used in another paper to develop a novel asymptotic extraction technique for the outer semi-infinite integral.

In this way, the inner surface of the cup becomes the phagosome membrane encaging the particle, and the outer surface remains an integral part of the plasma membrane surrounding the entire cell.