20f21c77-feec-43a6-abf7-16cba6d46e5c 32 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

x cumul u Φ u x - x int ≡ 2 x * 1 x 2^+ / ≡

u Φ 11 x 2^+ / :

x cumul u Φ u x - x int maple : 2 x * 1 x 2^+ /

Φ ∫ 2 x * 1 x 2^+ / :

x ∫ x Φ 2 1 sys

Φ ∫

∫

an image only

http://mathematica.stackexchange.com/questions/36961/paths-integrals-in-the-complex-plane

Convenient user col(data,n)

Alt+21 data n § data n col :

Alt+0143 data n  data n col :

Alt+20 data n ¶ data n col :

Alt+0126 data n ~ data n col :

Δ 0.01 1.787 1.785 5 1.519 1.52 10 1.307 1.31 20 1.002 1.004 30 0.798 0.797 40 0.653 0.651 50 0.547 0.545 60 0.467 0.465 70 0.404 0.404 80 0.355 0.355 90 0.315 0.316 100 0.282 0.285 123 mat :

Δ 2 § 1.787 1.519 1.307 1.002 0.798 0.653 0.547 0.467 0.404 0.355 0.315 0.282 121 mat

Δ 3  1.785 1.52 1.31 1.004 0.797 0.651 0.545 0.465 0.404 0.355 0.316 0.285 121 mat

Δ 1 ¶ 0.01 5102030405060708090100121 mat

Δ 2 ~ 1.787 1.519 1.307 1.002 0.798 0.653 0.547 0.467 0.404 0.355 0.315 0.282 121 mat

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Alt+0160

Alt+0187 data n » data n row :

Δ 3 » 10 1.307 1.31 13 mat

Alt+0151 data n — data n row :

Δ 5 — 30 0.798 0.797 13 mat

Curvilinear Integration

CDE Curvilinear Differential Element ≡

CDE x 2^y * dx * (x y + (dy * - (≡

on linear segment y x 2 / ≡ 0 x ≤ 2 ≤ dy 12 / dx * ≡ 31 line

u f x 2^x 2 / (* 12 / x x 2 / + (* - (:

u f x 02 int 0.5

x 2 ^ x 2 1 sys

on parabolic segment x 2 y 2^* ≡ 0 x ≤ 1 ≤ dx 4 y * dy * ≡ 31 line

u φ 2 y 2^* (2^4 * y 2^* 2 y 2^* y + (- :

u φ y 01 int 1.119048

1.119048 4742 / ≡

Contour Integration

t x 2 t cos * : t y 2 t sin * : t z t x i t y * + : 31 line

a 0 : b 2 π * : 12 mat

z f 1 z / : a complex function to evaluate ≡

t z the parametrized path in the z plane ≡

t f t z f t z t diff (* :

The integral of f(z) over the path ICntr t f t a b int : 6.283185 i *

This is 2π, not zero, indicating a pole within the circular path described by z(t).

z φ 1 z c - / :

more generally define

where

c abs 1 ≠

a 1 : b 1 : c 2 - : 31 line

General parametric equation x y Circle x 2^a 2^/ y 2^b 2^/ + 1 - :

Circle x φ 2 1 sys

Let f(z):=1/z and C the unit circle and one complete trigonometric rotation. The system can be parametrised by e^i*t, with t [0,2π]. The system becomes:

z f z # # int 1 e i t *^/ i * e i t *^* t 02 π * int ≡ i e i t * -^e i t *^* t 02 π * int * ≡ i 1 t 02 π * int * ≡

i 1 t 02 π * int * 6.283185 i *

Example: unit work done on this closed path

x y F x 2^x y * 12 mat ≡

y x 2^≡ t C t t 2^12 mat ≡ 0 t ≤ 1 ≡ t C t diff 12 t * 12 mat ≡ 14 mat t C F t C * t 2^t 3^12 mat 12 t * 12 mat * ≡ t 2^2 t 4^* + ≡ 21 line

C1 t 2^2 t 4^* + t 01 int : 0.733

1115 /

......

y x ≡ t C t t 12 mat ≡ 1 t ≥ 0 ≡ t C t diff 1112 mat ≡ 14 mat t C F t C * t 2^t 2^12 mat 1112 mat * ≡ 2 t 2^* ≡ 21 line

C2 2 t 2^* t 10 int : 0.667 -

2 - 3 /

...........

115 /

C1 C2 + 0.066667

.................

C t 01 0.01 range : j 1 t rows range c1 j el t j el (2^: c2 j el t j el : 21 line for t c1 augment t c2 augment 21 mat 31 line :

C 1 el C 2 el 1 1 . 15 black 1 5 mat 3 1 sys

C2 2 t 2^* t 10 int : 0.667 -

C1 t 2^2 t 4^* + t 01 int : 0.733

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