Fiboacci Notatios Traditioal ame Fiboacci umber Traditioal otatio F Ν Mathematica StadardForm otatio FiboacciΝ Primary defiitio 04..0.000.0 F Ν ΦΝ cosν Π Φ Ν Specific values Specialized values 04..03.000.0 F ; 04..03.000.0 F ; 04..03.00.0 F Φ Φ ; 04..03.006.0 F Φ Φ ; 04..03.007.0 F F ;
http://fuctios.wolfram.com 04..03.008.0 F Φ Φ ; Φ Φ Values at fied poits F 0 0 F F F 3 F 4 3 F F 6 8 04..03.0003.0 04..03.0004.0 04..03.000.0 04..03.0006.0 04..03.0007.0 04..03.0008.0 04..03.0009.0 04..03.000.0 F 7 3 04..03.00.0 F 8 04..03.00.0 F 9 34 04..03.003.0 F 0 Values at ifiities 04..03.004.0 F Geeral characteristics Domai ad aalyticity F Ν is a etire aalytical fuctio of Ν which is defied over the whole comple Ν-plae. 04..04.000.0 ΝF Ν Symmetries ad periodicities Parity
http://fuctios.wolfram.com 3 04..04.000.0 F F ; Mirror symmetry 04..04.0003.0 F Ν F Ν Periodicity No periodicity Poles ad essetial sigularities The fuctio F Ν has oly oe sigular poit at Ν. It is a essetial sigular poit. 04..04.0004.0 ig Ν F Ν, Brach poits The fuctio F Ν does ot have brach poits. 04..04.000.0 Ν F Ν Brach cuts The fuctio F Ν does ot have brach cuts. 04..04.0006.0 Ν F Ν Series represetatios Geeralized power series Epasios at geeric poit Ν Ν 0 For the fuctio itself 04..06.00.0 F Ν F Ν0 Ν 0 csch Ν 0 4 Ν 0 csch cosπ Ν 0 Π siπ Ν 0 Ν Ν 0 0 Π Π cosπ Ν 0 csch siπ Ν 0 Ν 0 csch F Ν0 Ν Ν 0 ; Ν Ν 0
http://fuctios.wolfram.com 4 04..06.006.0 F Ν F Ν0 Ν 0 csch Ν 0 4 Ν 0 csch cosπ Ν 0 Π siπ Ν 0 Ν Ν 0 0 Π Π cosπ Ν 0 csch siπ Ν 0 Ν 0 csch F Ν0 Ν Ν 0 OΝ Ν 0 3 04..06.007.0 F Ν F Ν 0 csch Ν 0 Ν 0 Π Ν 0 csch Π csch Π Ν 0 csch Π csch Ν Ν 0 04..06.008.0 F Ν F Ν0 OΝ Ν 0 Epasios at Ν 0 04..06.000.0 F Ν logφ Ν Π Ν log 3 Φ 3 Π logφ Ν3 ; Ν 0 04..06.009.0 F Ν logφ Ν Π Ν log 3 Φ 3 Π logφ Ν3 OΝ 4 04..06.000.0 F Ν csch Π csch Π csch Ν 04..06.0003.0 F Ν logφ Ν OΝ ; Ν 0 Asymptotic series epasios 04..06.0004.0 F Ν ΦΝ cosν Π Φ Ν ; Ν 04..06.000.0 Φ Ν ImΝ 0 ReΝ Π ImΝ 0 F Ν Ν ΠΝ csch csch Ν Π Ν ImΝ 0 Π ImΝ ReΝ 0 ImΝ 0 ReΝ Π ImΝ 0 ; Ν Φ Ν cosν Π Φ Ν True
http://fuctios.wolfram.com 04..06.00.0 F Ν ΦΝ ; Ν Other series represetatios F F 04..06.000.0 04..06.0006.0 04..06.0007.0 F ; ; ; 04..06.0008.0 F ; F 04..06.0009.0 04..06.000.0 ; F Π 4 ; 04..06.00.0 F ep ta ; F 04..06.00.0 04..06.003.0 F ; ; F 04..06.004.0 ; Itegral represetatios
http://fuctios.wolfram.com 6 O the real ais Of the direct fuctio F 3 04..07.000.0 0 Π 3 cost sitt ; Limit represetatios 04..09.000.0 F lim ma loglogd Μ m loglogμ Μ,,m ; d m d Σ 0 m d 0 m Σ 0 m Geeratig fuctios 04...000.0 t F t ; t t Differetial equatios Ordiary liear differetial equatios ad wrosias 04..3.000.0 w 3 Ν logφ w Ν Π log Φ w Ν logφ log Φ Π wν 0 ; wν c F Ν c L Ν c 3 Φ Ν siπ Ν Trasformatios Additio formulas 04..6.000.0 F m F F m F m F ; m 04..6.000.0 F m F L m F m L ; m 04..6.0003.0 F m F m ; m 04..6.0004.0 F m F m F F F m ; m 04..6.000.0 F m F m L F L m ; m
http://fuctios.wolfram.com 7 04..6.0006.0 F m F m ; m 04..6.0007.0 F m F m ; m 04..6.0008.0 F m3 F m ; m 04..6.0009.0 F Ν F Ν L Ν 04..6.000.0 F F F ; 04..6.007.0 F F F ; 0 Cassii's formula Multiple argumets 04..6.008.0 F Ν L Ν F Ν si Π Ν Φ Ν 04..6.009.0 F Ν F Ν F Ν F Ν F Ν Φ Ν si Π Ν 04..6.0030.0 F Ν F Ν F Ν F Ν F Ν Φ Ν si Π Ν 04..6.00.0 F F p F p F p F p ; p 04..6.00.0 F L F ; 04..6.003.0 F F ; 04..6.003.0 F Ν 3 F Ν F Ν 04..6.004.0 F m Ν L m F m Ν m F m Ν ; m
http://fuctios.wolfram.com 8 04..6.00.0 m m F m F F m F ; m m F m 04..6.006.0 m m m F L m ; m 04..6.007.0 m m F m F F m L 04..6.008.0 m m L m ; m F m m ; m 04..6.009.0 m m F m m m F m m ; m Products, sums, ad powers of the direct fuctio Products of the direct fuctio 04..6.000.0 F Ν F Ν F Ν cosν Π 04..6.00.0 F F m L m L m ; m Powers of the direct fuctio 04..6.00.0 F Ν F Ν F Ν cosν Π 04..6.003.0 F L ; 04..6.003.0 F 3 3 F F 3 ; 04..6.0033.0 F 4 4 F F 4 8 F F 4 6 ; 04..6.004.0 F 4 F F F F ;
http://fuctios.wolfram.com 9 04..6.00.0 F m m m m m F m m F m ; m Related trasformatios 04..6.006.0 F Ν L Ν L Ν Idetities Recurrece idetities Cosecutive eighbors 04..7.000.0 F Ν F Ν F Ν 04..7.000.0 F Ν F Ν F Ν 04..7.007.0 F Ν F Ν Φ Ν Φ 04..7.008.0 F Ν Φ F Ν Φ Ν Distat eighbors 04..7.0003.0 F Ν m U m 3 F mν m U m 3 F mν ; m 04..7.0004.0 F Ν m U m 3 F Νm m U m 3 F Νm ; m Fuctioal idetities Fuctioal equatios 04..7.009.0 wz wz wz ; wz c F z c L z Relatios of special id 04..7.0007.0 F Ν F Ν F Ν cosν Π 04..7.0008.0 F F l F l F F F l ; l 04..7.0009.0 F F F F F ;
http://fuctios.wolfram.com 0 04..7.000.0 F F m F m m F m ; m 04..7.00.0 4 F F F 3 F 4 F F 3 0 ; 04..7.00.0 F a F b F c b c a ; F cb F ab F b F ac F bc F c F ba F cb F a a b c a b a c b c 04..7.003.0 F Φ F ; 04..7.004.0 F gcdm, gcdf m, F ; m F m 04..7.00.0 m F F m F F F ; m 04..7.006.0 ta F ta F ta F ; Comple characteristics Real part 04..9.000.0 ReF y Φ Φ cosy logφ cosπ coshπ y cosy logφ siπ siy logφ sihπ y 04..9.0006.0 ReF y siπ siy csch sihπ y cosπ cosy csch coshπ y cosy csch Imagiary part 04..9.000.0 ImF y Φ Φ siy logφ cosπ coshπ y siy logφ cosy logφ siπ sihπ y 04..9.0007.0 ImF y cosπ coshπ y siy csch cosy csch siπ sihπ y siy csch
http://fuctios.wolfram.com Absolute value 04..9.0003.0 F y 0 Φ cosh Π y 4 Φ cosπ cos y logφ coshπ y Φ 4 sih Π y cos Π 4 Φ siπ si y logφ sihπ y F y 0 04..9.0008.0 3 4 cos Π cosh Π y 4 cosπ cos y csch coshπ y 4 siπ si y csch sihπ y Argumet 04..9.0004.0 argf y ta Φ Φ cosy logφ cosπ coshπ y cosy logφ siπ siy logφ sihπ y, Φ Φ siy logφ cosπ coshπ y siy logφ cosy logφ siπ sihπ y argf y 04..9.0009.0 ta siπ siy csch sihπ y cosπ cosy csch coshπ y cosy csch, cosπ coshπ y siy csch cosy csch siπ sihπ y siy csch Cojugate value 04..9.000.0 F y Φ Φ cosy logφ siy logφ siπ sihπ ysiy logφ cosy logφ cosπ coshπ y cosy logφ siy logφ 04..9.000.0 F y cosπ coshπ y cosy csch siy csch siπ siy csch cosy csch sihπ y cosy csch siy csch Sigum value
http://fuctios.wolfram.com 04..9.00.0 sgf y Φ cosπ coshπ y siy logφ cosy logφ Φ cosy logφ siy logφ siπ cosy logφ siy logφ sihπ y Φ cosh Π y 4 Φ cosπ cos y logφ coshπ y Φ 4 sih Π y cos Π 4 Φ siπ si y logφ sihπ y 04..9.00.0 sgf y siπ cosy csch siy csch sihπ y cosy csch siy csch cosπ coshπ y cosy csch siy csch 3 4 cos Π cosh Π y 4 cosπ cos y csch coshπ y 4 siπ si y csch sihπ y Differetiatio Low-order differetiatio F Ν 04..0.000.0 Ν ΦΝ Φ Ν logφ cosπ Ν logφ Π siπ Ν 04..0.000.0 F Ν Ν Φ Ν cosπ Ν Π log Φ logφ Φ Ν logφ Π siπ Ν Symbolic differetiatio 04..0.0003.0 F Ν Ν Φ Ν log Φ Φ Ν Π Ν logφ Π Π Ν logφ Π ; 04..0.0004.0 F Ν Ν F Ν log Φ ΦΝ cosπ Ν log Φ Π cos Π Ν log Φ ; Fractioal itegro-differetiatio 04..0.000.0 Α F Ν ΝΑ Ν Π csch Α ep Π csch ΝQΑ, Π csch Ν ep Π csch Ν Α Ν Ν Π csch Α QΑ, Π csch Ν Ν Α csch Α epν csch QΑ, Ν csch
http://fuctios.wolfram.com 3 Itegratio Idefiite itegratio Ivolvig oly oe direct fuctio F a Ν Ν 04...000.0 a 04...000.0 Φ a Ν logφ cosπ a Ν Π siπ a Ν log Φ Π logφ F Ν Ν Φ Ν logφ cosπ Ν Π siπ Ν log Φ Π logφ Ivolvig oe direct fuctio ad elemetary fuctios Ivolvig power fuctio Φ Ν Φ a Ν 04...0003.0 Ν Α F a Ν Ν Ν Α a Ν Α Α, a Ν csch csch Α a Ν Π csch Α Α, a Ν Π csch a Ν Π csch Α Α, a Ν Π csch 04...0004.0 ΝΑ Ν Α F Ν Ν Ν Α Α, Ν csch csch Α Ν Π csch Α Α, Ν Π csch Ν Π csch Α Α, Ν Π csch Itegral trasforms Laplace trasforms 04...000.0 t F t z z csch z csch z csch Π ; Rez logφ Summatio Fiite summatio 04..3.000.0 F F
http://fuctios.wolfram.com 4 04..3.00.0 F F ; 04..3.003.0 F F 3 ; 04..3.004.0 F z z z z F F ; z z 04..3.000.0 F q F pq z p F pq z p F qp z F pq z ; p q p z L p z 04..3.000.0 F F L F ; 04..3.00.0 F F F Ifiite summatio 04..3.0003.0 z F z z z F 4 04..3.0004.0 ϑ 0, 04..3.000.0 F F 3 04..3.0006.0 si Π F cos Π F 0 ; F F F F 04..3.0007.0 z F F F z z z z 4 z as a formal power series
http://fuctios.wolfram.com 04..3.00.0 F Φ F Φ Multiple sums 04..3.0008.0 m 0 m 0 m 0,j m j j F m j j j 0 j j ; 04..3.0009.0 p p j p j j F j F,p ; p F,p p F,p p F,p F, F Operatios Limit operatio F Ν lim Ν L Ν 04...000.0 F ΑΝ lim Φ Α Ν F Ν 04...000.0 04...0003.0 m F Ν lim Φ ; m Ν F mν F Ν Represetatios through more geeral fuctios Through hypergeometric fuctios Ivolvig F 04..6.000.0 F Ν Ν Π Ν cos F Ν, Ν ; 3 ; 4 si Π Ν F Ν, Ν ; ; 4 04..6.000.0 F Ν ΘΝ siν Π F Ν, Ν ; Ν; 4 ΘΝ siν Π cosν Π Ν F, Ν ; Ν ; 4 04..6.0003.0 Ν F Ν F, Ν ; Ν; 4 cosπ Ν Ν F, Ν ; Ν ; 4 ; Ν
http://fuctios.wolfram.com 6 F Ν 04..6.0004.0 Π Ν Ν siπ Ν cosπ Ν F Ν, Ν ; 3 ; 4 siπ Ν F Ν, Ν ; ; 4 04..6.000.0 F Ν 0 Π Ν Ν cosπ Ν siπ Ν F Ν, Ν; 3 ; 4 siπ Ν F Ν, Ν; ; 4 F Ν 0 04..6.0006.0 Π Ν siπ Ν F Ν, Ν ; ; 4 Ν siπ Ν cosπ Ν F Ν, Ν ; 3 ; 4 04..6.0007.0 F F, ; 3 ; ; 04..6.0008.0 F F, ; ; 4 ; 04..6.0009.0 F F, ; 3 ; 4 ; F 4 04..6.000.0 F, ; ; ; 04..6.00.0 F 4 F, ; ; ; 04..6.00.0 F F, ; ; 4 ; 04..6.003.0 F F, ; ; 4 ; F 04..6.004.0 F, ; ; 4 ; F 3 F 3 04..6.006.0 6 04..6.007.0 4 F 3 8 F 7 8, 3 8, 3 8 ; ; 9 ; ; 9 ; ;
http://fuctios.wolfram.com 7 04..6.008.0 F F, ; 3 ; 4 ; Ivolvig p F q 04..6.009.0 F 3F,, ; 3, ; 4 ; 04..6.000.0 F 4 3F,, ; 3, ; 4 ; Through Meijer G Classical cases for the direct fuctio itself siπ Ν F Ν Π 04..6.00.0, G 3,3 4 Ν, Ν, Ν 0,, Ν ; Ν F Ν 04..6.00.0 Ν Π G,, 4 Ν, Ν 0, Ν cosν Π Ν Π G,, 4 Ν, Ν 0, Ν ; Ν Geeralized cases for the direct fuctio itself siπ Ν F Ν Π 04..6.003.0, G 3,3, Ν, Ν, Ν 0,, Ν ; Ν Through other fuctios Ivolvig some hypergeometric-type fuctios 04..6.004.0 F Ν F Ν 04..6.00.0 F U ; Represetatios through equivalet fuctios With elemetary fuctios 04..7.000.0 F Ν Ν logw Π logw Ν Π logw Ν ; w
http://fuctios.wolfram.com 8 04..7.000.0 F Ν ΦΝ Ν ΠlogΦ Ν ΠlogΦ 04..7.0003.0 F Ν ΦΝ Ν ΠlogΦ Ν ΠlogΦ 04..7.0004.0 F Ν epν csch ep Ν csch cosπ Ν F Ν 04..7.000.0 Π Ν siπ Νcos Ν si cosπ Ν siπ Ν si Ν si 04..7.0006.0 F Ν cosπ Ν coshν logφ cosπ Ν sihν logφ 04..7.0007.0 F Ν si Π Ν si Ν csc Π Ν sihν csch 04..7.0008.0 F Ν Π Ν cosπ Ν siπ Ν si Ν si siπ Νcos Ν si F Ν 04..7.0009.0 Π Ν siπ Νcos Ν si siπ Ν cosπ Ν si Ν si 04..7.000.0 F si z siz ; z log Π With Lucas umbers 04..7.00.0 F Ν L Ν L Ν 04..7.00.0 F m L m L m L m L m L m ; m 04..7.003.0 F Ν L Ν L Ν Φ Ν si Π Ν L Ν L Ν
http://fuctios.wolfram.com 9 Other idetities Idetities ivolvig determiats 04..3.000.0 F if l if l 0 else l Theorems Zecedorf theorem Every positive iteger ca be decomposed i a uique way as a sum of Fiboacci umbers, such that o two of these umbers are cosecutive i the Fiboacci sequece. Fiboacci substitutio After actig o A times with the Fiboacci substitutio A A B, B A the resultig sequece cotais F As ad F Bs. A trascedetal umber F is a trascedetal umber. The umbers of primary ad secodary spirals i the positios of leaves The umbers of primary ad secodary spirals i the positios of leaves or scales alog a plat stem are early always two cosecutive Fiboacci umbers. Hirmer's cojecture The umber of the largest set of oitersectig circles arraged alog the circumferece of a give circle ad agle Π GoldeRatio betwee cosecutive midpoits is give by the Fiboacci umbers F. History J. Kepler (608) A. Girard (634); R. Simpso (73) É. Léger (837) É. Lucas (870, 876 880) G.H. Hardy ad E.M. Wright (938)
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