Tazama chanzo cha Ufafanuzi mfululizo

[[File:Taylor cos.gif|alt=Approximation of cosine by a Taylor series|thumb|Uhaishaji unaoonyesha [[husisho kosini]] likikadiriwa na vifupisho vya [[Mfululizo wa Maclaurin|mfululizo wake wa Maclaurin]].]]
Katika [[hisabati]], '''ufafanuzi mfululizo''' ni mbinu ambayo huonyesha [[husisho]] kama jumla [[Usokomo|isiyokoma]] (au [[mfululizo]]) ya mahusisho rahisi zaidi. Ni mbinu ya kukokotoa mahusisho yasiyoweza kuonyeshwa kwa [[Tendo (hisabati)|matendo]] ya msingi tu ([[ujumlishaji]], [[utoaji]], [[uzidishaji]], na [[ugawanyaji]]).<ref>{{Cite book|url=https://books.google.com/books?id=WGHWBoX89doC&q=%22series+expansion%22+AND+%22approximation%22|title=Numerical Methods for Special Functions|last1=Gil|first1=Amparo|last2=Segura|first2=Javier|last3=Temme|first3=Nico M.|date=2007-01-01|publisher=SIAM|isbn=978-0-89871-782-2|language=en}}</ref>

Mara nyingi, mfululizo tokeo unaweza kufupishwa hadi idadi ya vipengele vikomavyo, hivyo kutoa [[kadirio]] la husisho hilo. Kiasi vipengele vya mfululizo vinavyojumulishwa, ndivyo kadirio linavyokuwa sahihi zaidi. Mara nyingi, upungufu unaotokea kati ya kadirio na thamani halisi (yaani, jumla ya vipengele vilivyotelekezwa) unaweza kuelezwa kwa [[mwandiko wa O kubwa]] (tazama pia: [[ufafanuzi wa kiugo]]).<ref>{{Cite web|date=2013-11-07|title=Series and Expansions|url=https://math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Series_and_Expansions|access-date=2021-12-24|website=Mathematics LibreTexts|language=en}}</ref>

== Aina za ufafanuzi mfululizo ==
=== Mfululizo wa Taylor ===
{{Main|Mfululizo wa Taylor}}
Mfululizo wa Taylor ni [[mfululizo wa vipeo]] unaotokana na [[Kinyambuo (hisabati)|vinyambuo]] kwenye nukta moja.<ref name=":1">{{Cite web|date=27 Desemba 2013|title=Taylor series - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Taylor_series|website=encyclopediaofmath.org|accessdate=18 Septemba 2024}}</ref> Kihususa, kama husisho <math>f: U\to\R</math> likitenguka kiusokomo kuzunguka nukta <math>x_0</math>, hivyo mfululizo wa Taylor wa <math>f</math> kuzunguka nukta hiyo watolewa na:

<math>\sum_{n=0}^{\infty}\frac{f^{(n)}(x_0)}{n!}(x - x_0)^n</math>

ikiwa desturi <math>0^0 := 1</math>.<ref name=":1" /><ref name=":2">{{Cite book |last1=Edwards |first1=C. Henry |title=Elementary Differential Equations with Boundary Value Problems |last2=Penney |first2=David E. |year=2008 |isbn=978-0-13-600613-8 |pages=196 |publisher=Pearson/Prentice Hall |language=en}}</ref> [[Mfululizo wa Maclaurin]] wa <math>f</math> ni mfululizo wake wa Taylor kuzunguka <math>x_0 = 0</math>.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Maclaurin Series|url=https://mathworld.wolfram.com/|access-date=2022-03-22|website=mathworld.wolfram.com|language=en}}</ref><ref name=":2" />  

=== Mfululizo wa Laurent ===
{{Main|Mfululizo wa Laurent}}
Mfululizo wa Laurent ni [[jumuisho]] la mfululizo wa Taylor, ukiruhusu vipengele vyenye vipeo hasi; mfululizo una umbo wa  <math display="inline">\sum_{k = -\infty}^{\infty} c_k (z - a)^k</math> na hukutana kuelekea [[Kipete (hisabati)|kipete]].<ref>{{Cite web|title=Laurent series - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Laurent_series|access-date=2022-03-22|website=encyclopediaofmath.org}}</ref> Hasa mfululizo wa Laurent unaweza kutumiwa ili kuchunguza mwenendo wa [[husisho changamano]] karibu na [[Shani (hisabati)|shani]] ukifikiriwa ufafanuzi mfululizo kwenye kipete chenye shani ya kitovuni.

=== Mfululizo wa Dirichlet ===
{{Main|Mfululizo wa Dirichlet}}
Mfululizo wa Dirichet una umbo <math display="inline">\sum_{n = 1}^{\infty} a_ne^{-\lambda_n s}.</math>

=== Mfululizo wa Fourier ===
{{Main|Mfululizo wa Fourier}}

Mfululizo wa Fourier ni ufafanuzi wa [[Husisho la kurudiarudia|mahusisho ya kurudiarudia]] kama jumla ya [[Husisho sini|sini]] na [[Husisho kosini|kosini]].<ref name=":42">{{Cite web|title=Fourier series - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Fourier_series|access-date=2022-03-22|website=encyclopediaofmath.org|language=en}}</ref> Mfululizo wa Fourier wa husisho <math>f(t)</math> lenye kipindi cha <math>2L</math> watolewa na [[Uonyesho (hisabati)|uonyesho]]:

<math>a_0 + \sum_{n = 1}^{\infty} \left[a_n\cos\left(\frac{n\pi t}{L}\right) + b_n\sin\left(\frac{n\pi t}{L}\right)\right]</math>

ambapo [[Kizigeu|vizigeu]] vyatolewa na [[fomula]]:<ref>{{Cite book|title=Elementary Differential Equations with Boundary Value Problems|last1=Edwards|first1=C. Henry|last2=Penney|first2=David E.|publisher=Pearson/Prentice Hall|year=2008|isbn=978-0-13-600613-8|pages=558, 564|language=en}}</ref>

<math>\begin{align}
a_n &:= \frac{1}{L}\int_{-L}^L f(t)\cos\left(\frac{n\pi t}{L}\right)dt, \\
b_n &:= \frac{1}{L}\int_{-L}^L f(t)\sin\left(\frac{n\pi t}{L}\right)dt.
\end{align}</math>

==Mifano==
Yafuatayo ni mfululizo wa Taylor wa <math>e^x</math>: <math>e^x=\sum^{\infty}_{n=0}\frac{x^n}{n!}= 1 + x + \frac{x^2}{2} + \frac{x^3}{6}...</math><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Exponential Function|url=https://mathworld.wolfram.com/ExponentialFunction.html|access-date=2021-08-12|website=mathworld.wolfram.com|language=en}}</ref><ref>{{Cite web|date=5 Juni 2020|title=Exponential function - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Exponential_function|access-date=18 Septemba 2024|website=encyclopediaofmath.org|language=en}}</ref>

Mfululizo wa Dirichlet wa [[husisho zeta la Reimann]] ni <math>\zeta(s) := \sum_{n = 1}^{\infty} \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \cdots</math><ref name=":3">{{Cite web|date=26 Januari 2022|title=Dirichlet series - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Dirichlet_series|access-date=18 Septemba 2024|website=encyclopediaofmath.org}}</ref>

== Marejeo ==
{{marejeo}}
[[Jamii:Hisabati]]