WikiDer > Trigonometrik integral

Si (x) (ko'k) va Ci (x) (yashil) bir xil uchastkada chizilgan.

Yilda matematika, trigonometrik integrallar a oila ning integrallar jalb qilish trigonometrik funktsiyalar.

Sinus integral

Uchastka Si (x) uchun 0 ≤ x ≤ 8 π.

Turli xil sinus ajralmas ta'riflar

${ displaystyle operator nomi {Si} (x) = int _ {0} ^ {x} { frac { sin t} {t}} , dt}$

${ displaystyle operator nomi {si} (x) = - int _ {x} ^ { infty} { frac { sin t} {t}} , dt ~.}$

Integrand ekanligini unutmang^{gunoh x}⁄ _x bo'ladi sinc funktsiyasi, shuningdek, nol sferik Bessel funktsiyasi.Bundan beri samimiy bu hatto butun funktsiya (holomorfik butun murakkab tekislikda), Si butun, g'alati va uning ta'rifidagi integralni olish mumkin har qanday yo'l so'nggi nuqtalarni ulash.

Ta'rifga ko'ra, Si (x) bo'ladi antivivativ ning gunoh x / x uning qiymati nolga teng x = 0va si (x) qiymati antidivivdir, uning qiymati nolga teng x = ∞. Ularning farqlari Dirichlet integrali,

${ displaystyle operator nomi {Si} (x) - operator nomi {si} (x) = int _ {0} ^ { infty} { frac { sin t} {t}} , dt = { frac { pi} {2}} quad { text {or}} quad operatorname {Si} (x) = { frac { pi} {2}} + operatorname {si} (x) ~ .}$

Yilda signallarni qayta ishlash, sinus integral sababining tebranishlari overshoot va qo'ng'iroq qilayotgan buyumlar dan foydalanganda sinc filtriva chastota domeni kesilgan sinc filtrini a sifatida ishlatsangiz qo'ng'iroq past o'tkazgichli filtr.

Bog'liq Gibbs hodisasi: Agar sinus integral integral sifatida qabul qilinsa konversiya bilan sinc funksiyasining og'ir funksiya, bu qisqartirishga to'g'ri keladi Fourier seriyasi, bu Gibbs hodisasining sababi.

Kosinus integrali

Uchastka Ci (x) uchun 0 < x ≤ 8π .

Turli xil kosinus ajralmas ta'riflar

${ displaystyle operator nomi {Cin} (x) = int _ {0} ^ {x} { frac {1- cos t} {t}} operator nomi {d} t ~,}$

${ displaystyle operator nomi {Ci} (x) = - int _ {x} ^ { infty} { frac { cos t} {t}} operator nomi {d} t = gamma + ln x- int _ {0} ^ {x} { frac {1- cos t} {t}} operatorname {d} t qquad ~ { text {for}} ~ left | operatorname {Arg} ( x) right | < pi ~,}$

qayerda γ ≈ 0.57721566 ... bu Eyler-Maskeroni doimiysi. Ba'zi matnlardan foydalaniladi ci o'rniga Salom.

Ci (x) ning antiderivatividir cos x / x (bu yo'qoladi ${ displaystyle x to infty}$). Ikki ta'rif bir-biriga bog'liqdir

${ displaystyle operator nomi {Ci} (x) = gamma + ln x- operator nomi {Cin} (x) ~.}$

Cin bu hatto, butun funktsiya. Shu sababli, ba'zi matnlarda muomala qilinadi Cin asosiy funktsiya sifatida va hosil qiladi Salom xususida Cin.

Giperbolik sinus integral

The giperbolik sinus integral sifatida belgilanadi

${ displaystyle operator nomi {Shi} (x) = int _ {0} ^ {x} { frac { sinh (t)} {t}} , dt.}$

Bu oddiy sinus integral bilan bog'liq

${ displaystyle operator nomi {Si} (ix) = i operator nomi {Shi} (x).}$

Giperbolik kosinus integrali

The giperbolik kosinus ajralmas hisoblanadi

${ displaystyle operator nomi {Chi} (x) = gamma + ln x + int _ {0} ^ {x} { frac {; cosh t-1 ;} {t}} operator nomi {d } t qquad ~ { text {for}} ~ left | operatorname {Arg} (x) right | < pi ~,}$

qayerda ${ displaystyle gamma}$ bo'ladi Eyler-Maskeroni doimiysi.

Uning ketma-ket kengayishi bor

${ displaystyle operator nomi {Chi} (x) = gamma + ln (x) + { frac {x ^ {2}} {4}} + { frac {x ^ {4}} {96}} + { frac {x ^ {6}} {4320}} + { frac {x ^ {8}} {322560}} + { frac {x ^ {10}} {36288000}} + O (x ^ {12}).}$

Yordamchi funktsiyalar

Trigonometrik integrallarni "yordamchi funktsiyalar" deb atash mumkin

${ displaystyle { begin {array} {rcl} f (x) & equiv & int _ {0} ^ { infty} { frac { sin (t)} {t + x}} mathrm { d} t & = & int _ {0} ^ { infty} { frac {e ^ {- xt}} {t ^ {2} +1}} mathrm {d} t & = & quad operatorname { Ci} (x) sin (x) + chap [{ frac { pi} {2}} - operatorname {Si} (x) right] cos (x) ~, qquad { text { va}} g (x) & equiv & int _ {0} ^ { infty} { frac { cos (t)} {t + x}} mathrm {d} t & = & int _ {0} ^ { infty} { frac {te ^ {- xt}} {t ^ {2} +1}} mathrm {d} t & = & - operator nomi {Ci} (x) cos ( x) + left [{ frac { pi} {2}} - operatorname {Si} (x) right] sin (x) ~. end {array}}}$

Ushbu funktsiyalar yordamida trigonometrik integrallar qayta ifodalanishi mumkin (qarang: Abramovits va Shtegun, p. 232)

${ displaystyle { begin {array} {rcl} { frac { pi} {2}} - operatorname {Si} (x) = - operatorname {si} (x) & = & f (x) cos (x) + g (x) sin (x) ~, qquad { text {and}} operatorname {Ci} (x) & = & f (x) sin (x) -g (x)) cos (x) ~. end {massiv}}}$

Nilsen spirali

Nilsen spirali.

The spiral ning parametrik chizmasi bilan hosil qilingan si, ci Nilsen spirali sifatida tanilgan.

${ displaystyle x (t) = a times operatorname {ci} (t)}$
${ displaystyle y (t) = a times operatorname {si} (t)}$

Spiral bilan chambarchas bog'liq Frenel integrallari va Eyler spirali. Nilsen spirali ko'rishni qayta ishlash, yo'l va yo'l qurilishida va boshqa sohalarda qo'llaniladigan dasturlarga ega.^{[iqtibos kerak]}

Kengayish

Trigonometrik integrallarni baholash uchun argument doirasiga qarab har xil kengayishlardan foydalanish mumkin.

Asimptotik seriya (katta argument uchun)

${ displaystyle operator nomi {Si} (x) sim { frac { pi} {2}} - { frac { cos x} {x}} chap (1 - { frac {2!} { x ^ {2}}} + { frac {4!} {x ^ {4}}} - { frac {6!} {x ^ {6}}} cdots right) - { frac { sin x} {x}} left ({ frac {1} {x}} - { frac {3!} {x ^ {3}}} + { frac {5!} {x ^ {5} }} - { frac {7!} {x ^ {7}}} cdots right)}$

${ displaystyle operator nomi {Ci} (x) sim { frac { sin x} {x}} left (1 - { frac {2!} {x ^ {2}}} + { frac { 4!} {X ^ {4}}} - { frac {6!} {X ^ {6}}} cdots right) - { frac { cos x} {x}} chap ({ frac {1} {x}} - { frac {3!} {x ^ {3}}} + { frac {5!} {x ^ {5}}} - { frac {7!} {x ^ {7}}} cdots o'ng) ~.}$

Ushbu seriyalar asimptotik va turlicha, ammo taxmin qilish va hatto aniq baholash uchun ishlatilishi mumkin ℜ (x) ≫ 1.

Konvergent seriyali

${ displaystyle operator nomi {Si} (x) = sum _ {n = 0} ^ { infty} { frac {(-1) ^ {n} x ^ {2n + 1}} {(2n + 1) ) (2n + 1)!}} = X - { frac {x ^ {3}} {3! Cdot 3}} + { frac {x ^ {5}} {5! Cdot 5}} - { frac {x ^ {7}} {7! cdot 7}} pm cdots}$

${ displaystyle operator nomi {Ci} (x) = gamma + ln x + sum _ {n = 1} ^ { infty} { frac {(-1) ^ {n} x ^ {2n}} { 2n (2n)!}} = Gamma + ln x - { frac {x ^ {2}} {2! Cdot 2}} + { frac {x ^ {4}} {4! Cdot 4 }} mp cdots}$

Ushbu ketma-ketliklar har qanday kompleksda yaqinlashadi x, garchi uchun |x| ≫ 1, ketma-ketlik dastlab asta sekin birlashadi va yuqori aniqlik uchun ko'p shartlarni talab qiladi.

Ketma-ket kengayishni keltirib chiqarish

${ displaystyle sin , x = x - { frac {x ^ {3}} {3!}} + { frac {x ^ {5}} {5!}} - { frac {x ^ { 7}} {7!}} + { Frac {x ^ {9}} {9!}} - { frac {x ^ {11}} {11!}} + , ...}$(Maclaurin seriyasining kengayishi)

${ displaystyle { frac { sin , x} {x}} = 1 - { frac {x ^ {2}} {3!}} + { frac {x ^ {4}} {5!} } - { frac {x ^ {6}} {7!}} + { frac {x ^ {8}} {9!}} - { frac {x ^ {10}} {11!}} + , ...}$

${ displaystyle Shuning int { frac { sin , x} {x}} dx = x - { frac {x ^ {3}} {3! cdot 3}} + { frac {x ^ {5}} {5! Cdot 5}} - { frac {x ^ {7}} {7! Cdot 7}} + { frac {x ^ {9}} {9! Cdot 9}} - { frac {x ^ {11}} {11! cdot 11}} + , ...}$

Xayoliy argumentning eksponent integrali bilan bog'liqligi

Funktsiya

$${ displaystyle operator nomi {E} _ {1} (z) = int _ {1} ^ { infty} { frac { exp (-zt)} {t}} , dt qquad ~ { matn {for}} ~ ~ Re (z) geq 0}$$

deyiladi eksponent integral. Bu bilan chambarchas bog'liq Si va Salom,

$${ displaystyle operator nomi {E} _ {1} (ix) = i chap (- { frac { pi} {2}} + operator nomi {Si} (x) o'ng) - operator nomi {Ci} (x) = i operator nomi {si} (x) - operator nomi {ci} (x) qquad ~ { text {for}} ~ x> 0 ~.}$$

Har bir tegishli funktsiya argumentning salbiy qiymatlari kesimidan tashqari analitik bo'lgani uchun, munosabatlarning amal qilish doirasi (Ushbu diapazondan tashqari, butun son omillari bo'lgan qo'shimcha atamalar) ga kengaytirilishi kerak. π ifodada paydo bo'ladi.)

Umumlashtirilgan integral-eksponent funktsiyani xayoliy argumentlari holatlari

$${ displaystyle int _ {1} ^ { infty} cos (ax) { frac { ln x} {x}} , dx = - { frac { pi ^ {2}} {24} } + gamma chap ({ frac { gamma} {2}} + ln a o'ng) + { frac { ln ^ {2} a} {2}} + sum _ {n geq 1} { frac {(-a ^ {2}) ^ {n}} {(2n)! (2n) ^ {2}}} ~,}$$

bu haqiqiy qismi

$${ displaystyle int _ {1} ^ { infty} e ^ {iax} { frac { ln x} {x}} , operator nomi {d} x = - { frac { pi ^ {2 }} {24}} + gamma chap ({ frac { gamma} {2}} + ln a o'ng) + { frac { ln ^ {2} a} {2}} - { frac { pi} {2}} i chap ( gamma + ln a right) + sum _ {n geq 1} { frac {(ia) ^ {n}} {n! n ^ { 2}}} ~.}$$

Xuddi shunday

$${ displaystyle int _ {1} ^ { infty} e ^ {iax} { frac { ln x} {x ^ {2}}} , operatorname {d} x = 1 + ia left [ - { frac {; pi ^ {2}} {24}} + gamma chap ({ frac { gamma} {2}} + ln a-1 o'ng) + { frac { ln ^ {2} a} {2}} - ln a + 1 right] + { frac { pi a} {2}} { Bigl (} gamma + ln a-1 { Bigr) } + sum _ {n geq 1} { frac {(ia) ^ {n + 1}} {(n + 1)! n ^ {2}}} ~.}$$

Samarali baholash

Padé taxminiy vositalari konvergent Teylor qatori kichik argumentlar uchun funktsiyalarni baholashning samarali usulini beradi. Rowe va boshqalar tomonidan berilgan quyidagi formulalar. (2015),^[1] ga nisbatan aniqroq 10⁻¹⁶ uchun 0 ≤ x ≤ 4,

${ displaystyle { begin {array} {rcl} operatorname {Si} (x) & approx & x cdot left ({ frac { begin {array} {l} 1-4.54393409816329991 cdot 10 ^ {- 2} cdot x ^ {2} +1.15457225751016682 cdot 10 ^ {- 3} cdot x ^ {4} -1.41018536821330254 cdot 10 ^ {- 5} cdot x ^ {6} ~~~ + 9.43280809438713025 cdot 10 ^ {- 8} cdot x ^ {8} -3.53201978997168357 cdot 10 ^ {- 10} cdot x ^ {10} +7.08240282274875911 cdot 10 ^ {- 13} cdot x ^ {12} ~~~ -6.05338212010422477 cdot 10 ^ {- 16} cdot x ^ {14} end {array}} { begin {array} {l} 1 + 1.01162145739225565 cdot 10 ^ {- 2} cdot x ^ {2} +4.99175116169755106 cdot 10 ^ {- 5} cdot x ^ {4} +1.55654986308745614 cdot 10 ^ {- 7} cdot x ^ {6} ~~~ + 3.28067571055789734 cdot 10 ^ { -10} cdot x ^ {8} +4.5049097575386581 cdot 10 ^ {- 13} cdot x ^ {10} +3.21107051193712168 cdot 10 ^ {- 16} cdot x ^ {12} end {array}} } o'ng) & ~ & operator nomi {Ci} (x) & approx & gamma + ln (x) + && x ^ {2} cdot chap ({ frac { begin) {array} {l} -0.25 + 7.51851524438898291 cdot 10 ^ {- 3} cdot x ^ {2} -1.27528342240267686 cdot 10 ^ {- 4} cdot x ^ {4} +1.05297363846239184 cdot 10 ^ {- 6} cdot x ^ {6} ~~~ -4.68889 508144848019 cdot 10 ^ {- 9} cdot x ^ {8} +1.06480802891189243 cdot 10 ^ {- 11} cdot x ^ {10} -9.93728488857585407 cdot 10 ^ {- 15} cdot x ^ {12} end {array}} { begin {array} {l} 1 + 1.1592605689110735 cdot 10 ^ {- 2} cdot x ^ {2} +6.72126800814254432 cdot 10 ^ {- 5} cdot x ^ { 4} +2.55533277086129636 cdot 10 ^ {- 7} cdot x ^ {6} ~~~ + 6.97071295760958946 cdot 10 ^ {- 10} cdot x ^ {8} +1.38536352772778619 cdot 10 ^ {- 12 } cdot x ^ {10} +1.89106054713059759 cdot 10 ^ {- 15} cdot x ^ {12} ~~~ + 1.39759616731376855 cdot 10 ^ {- 18} cdot x ^ {14} end {array}}} right) end {array}}}$

Integrallarni bilvosita yordamchi funktsiyalar orqali baholash mumkin ${ displaystyle f (x)}$ va ${ displaystyle g (x)}$tomonidan belgilanadigan

$${ displaystyle operator nomi {Si} (x) = { frac { pi} {2}} - f (x) cos (x) -g (x) sin (x)}$$		$${ displaystyle operator nomi {Ci} (x) = f (x) sin (x) -g (x) cos (x)}$$
yoki unga teng ravishda
$${ displaystyle f (x) equiv left [{ frac { pi} {2}} - operatorname {Si} (x) right] cos (x) + operatorname {Ci} (x) gunoh (x)}$$		$${ displaystyle g (x) equiv left [{ frac { pi} {2}} - operatorname {Si} (x) right] sin (x) - operatorname {Ci} (x) cos (x)}$$

Uchun ${ displaystyle x geq 4}$ The Padening ratsional funktsiyalari taxminan quyida keltirilgan ${ displaystyle f (x)}$ va ${ displaystyle g (x)}$ 10 dan kam xato bilan⁻¹⁶:^[1]

${ displaystyle { begin {array} {rcl} f (x) & approx & { dfrac {1} {x}} cdot left ({ frac { begin {array} {l} 1 + 7.44437068161936700618 cdot 10 ^ {2} cdot x ^ {- 2} +1.96396372895146869801 cdot 10 ^ {5} cdot x ^ {- 4} +2.37750310125431834034 cdot 10 ^ {7} cdot x ^ {- 6} ~~~ + 1.43073403821274636888 cdot 10 ^ {9} cdot x ^ {- 8} +4.33736238870432522765 cdot 10 ^ {10} cdot x ^ {- 10} +6.40533830574022022911 cdot 10 ^ {11} d ^ {- 12} ~~~ + 4.20968180571076940208 cdot 10 ^ {12} cdot x ^ {- 14} +1.00795182980368574617 cdot 10 ^ {13} cdot x ^ {- 16} +4.94816688199951963482 cdot 10 ^ {12} cdot x ^ {- 18} ~~~ -4.94701168645415959931 cdot 10 ^ {11} cdot x ^ {- 20} end {array}} { begin {array} {l} 1+ 7.46437068161927678031 cdot 10 ^ {2} cdot x ^ {- 2} +1.97865247031583951450 cdot 10 ^ {5} cdot x ^ {- 4} +2.41535670165126845144 cdot 10 ^ {7} cdot x ^ {- 6} ~~~ + 1.47478952192985464958 cdot 10 ^ {9} cdot x ^ {- 8} +4.58595115847765779830 cdot 10 ^ {10} cdot x ^ {- 10} +7.08501308149515401563 cdot 10 ^ {11} cdot x ^ {- 12} ~~~ + 5.06084464593475076774 cdot 10 ^ {12} cdot x ^ {- 14} +1.43468549171581016479 cdot 10 ^ {13} cdot x ^ {- 16} +1.11535493509914254097 cdot 10 ^ {13} cdot x ^ {- 18} end {array}}} right) && g (x) & approx & { dfrac {1} {x ^ {2}}} cdot left ({ frac { begin {array} {l} 1 + 8.1359520115168615 cdot 10 ^ {2} cdot x ^ {- 2} +2.35239181626478200 cdot 10 ^ {5} cdot x ^ {- 4} +3.12557570795778731 cdot 10 ^ {7} cdot x ^ {- 6} ~~~ + 2.06297595146763354 cdot 10 ^ {9} cdot x ^ {- 8} +6.83052205423625007 cdot 10 ^ {10} cdot x ^ {- 10} +1.09049528450362786 cdot 10 ^ {12} cdot x ^ {- 12} ~~~ + 7.57664583257834349 cdot 10 ^ {12} cdot x ^ {- 14} +1.81004487464664575 cdot 10 ^ {13} cdot x ^ {- 16} +6.43291613143049485 cdot 10 ^ {12} cdot x ^ {- 18} ~~~ -1.36517137670871689 cdot 10 ^ {12} cdot x ^ {- 20} end {array}} { begin {array} {l} 1 + 8.19595201151451564 cdot 10 ^ {2} cdot x ^ { -2} +2.40036752835578777 cdot 10 ^ {5} cdot x ^ {- 4} +3.26026661647090822 cdot 10 ^ {7} cdot x ^ {- 6} ~~~ + 2.23355543278099360 cdot 10 ^ {9 } cdot x ^ {- 8} +7.87465017341829930 cdot 10 ^ {10} cdot x ^ {- 10} +1.39866710696414565 cdot 10 ^ {12} cdot x ^ {- 12} ~~~ + 1.17164723371736605 cdot 10 ^ {13} cdot x ^ {- 14} +4.01839087307 656620 cdot 10 ^ {13} cdot x ^ {- 16} +3.99653257887490811 cdot 10 ^ {13} cdot x ^ {- 18} end {array}}} right) end {array} }}$

Shuningdek qarang

• Logaritmik integral
• Tans funktsiyasi
• Tanhc funktsiyasi
• Sinhc funktsiyasi
• Coshc funktsiyasi

Adabiyotlar

• Abramovits, Milton; Stegun, Irene Ann, eds. (1983) [1964 yil iyun]. "5-bob". Matematik funktsiyalar uchun formulalar, grafikalar va matematik jadvallar bilan qo'llanma. Amaliy matematika seriyasi. 55 (To'qqizinchi o'ninchi asl nashrning tuzatishlar bilan qo'shimcha tuzatishlar bilan qayta nashr etilishi (1972 yil dekabr); birinchi nashr). Vashington Kolumbiyasi; Nyu-York: Amerika Qo'shma Shtatlari Savdo vazirligi, Milliy standartlar byurosi; Dover nashrlari. p. 231. ISBN 978-0-486-61272-0. LCCN 64-60036. JANOB 0167642. LCCN 65-12253.

Qo'shimcha o'qish

• Mathar, R.J. (2009). "Tebranuvchi integralni exp ustidan sonli baholash (menπx)·x^1/x 1 dan ∞ gacha ". B ilova. arXiv:0912.3844 [math.CA].
• Press, W.H .; Teukolskiy, S.A .; Vetterling, Vt .; Flannery, B.P. (2007). "6.8.2-bo'lim - kosinus va sinus integrallari". Raqamli retseptlar: Ilmiy hisoblash san'ati (3-nashr). Nyu-York: Kembrij universiteti matbuoti. ISBN 978-0-521-88068-8.
• Qichqiriq, Dan. "Sine Integral Taylor seriyasining isboti" (PDF). Differentsial tenglamalardan farqli tenglamalar.
• Temme, NM (2010), "Eksponensial, logaritmik, sinusli va kosinaviy integrallar", yilda Olver, Frank V. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Klark, Charlz V. (tahr.), NIST Matematik funktsiyalar bo'yicha qo'llanma, Kembrij universiteti matbuoti, ISBN 978-0-521-19225-5, JANOB 2723248

Tashqi havolalar

• http://mathworld.wolfram.com/SineIntegral.html
• "Integral sinus", Matematika entsiklopediyasi, EMS Press, 2001 [1994]
• "Ajralmas kosinus", Matematika entsiklopediyasi, EMS Press, 2001 [1994]