Upload File

π!: extending the factorial function

1
π!: Extending the Factorial Function M Cap Khoury Abstract for 7 October 2010 What should we mean by π!? The gamma function Γ(s) is a well-known way to extend n! to (almost) all the complex numbers. But is it fair to call Γ(s), as many people do, the extension of n!? It isn't the only function which agrees (after a shift) with n! on the natural numbers, not even the only meromorphic one. We will review Γ(s) and the uniqueness properties it does (and doesn't) have; we will also introduce the Hadamard function and, time permitting, other interesting extensions of n!. For dessert, we will present Manjul Bhargava's wonderfully elegant generalized factorial construction. This talk is aimed primarily at upper- level undergraduate math majors but should have something for everyone.

Upload: others

Post on 26-Mar-2022

2 views

Category:

Documents


0 download

Report

TRANSCRIPT

Page 1: π!: Extending the Factorial Function

π!: Extending the Factorial Function

M Cap KhouryAbstract for 7 October 2010

What should we mean by π!? The gamma function Γ(s) is a well-known way to extend n! to (almost) all the complex numbers. But is it fair to call Γ(s), as many people do, the extension of n!? It isn't the only function which agrees (after a shift) with n! on the natural numbers, not even the only meromorphic one. We will review Γ(s) and the uniqueness properties it does (and doesn't) have; we will also introduce the Hadamard function and, time permitting, other interesting extensions of n!. For dessert, we will present Manjul Bhargava's wonderfully elegant generalized factorial construction. This talk is aimed primarily at upper-level undergraduate math majors but should have something for everyone.

Factorial BG ANOVA Psy 420 Ainsworth. Topics in Factorial Designs Factorial? Crossing and Nesting Assumptions…

Extending CORBA Interfaces with π-calculus for Protocol Compatibility

15. Multivariable QFT Design (Part 2)pcsl.ecs.umass.edu/cbs/handouts/ch15_MIMO_II_handout.pdf · cccc cc ⎛⎞ππ−ππ+π ⎛π+π ⎞ =π+⎜⎟+ =π+⎜+⎟ ⎝⎠π+ ⎝π+

3. Factorial Experiments (Ch.5. Factorial Experiments)

Designing experiments with several factors Outlines: Factorial Experiments Two-factor factorial experiment General factorial experiments 2 k factorial

Dynamic nonlinear elasticity in geomaterials€¦ · DYNAMIC NONLINEAR ELASTICITY IN GEOMATERIALS 3 x σ = 0 σ = 1-π-π π π x σ = _ σ = 4-π-π π π 2 π x σ >> 1-π π Amplitude

filotis · ∂§§∏¡π∫∏ ¢∏ª√∫ƒ∞∆π∞ À¶√Àƒ°∂π√ ∂£¡π∫∏™ ¶∞π¢∂π∞™ ∫∞𠣃∏™∫∂Àª∞∆ø¡ ¢π∂À£À

Review of Factorial Designs 5 terms necessary to understand factorial designs 5 patterns of factorial results for a 2x2 factorial designs Descriptive &

2k Factorial Factorial Designs - cse.wustl.edujain/cse567-08/ftp/k_172kd.pdf · 2k Factorial Designs, 22 Factorial Designs, Model, Computation of Effects, Sign Table Method, Allocation

Factorial Experiments: - Blocking, -Confounding, and -Fractional Factorial Designs

Dalitz analysis of the decay * B + →π + π - π +

Dynamics of η → π 0 π 0 π 0

Design-plots for factorial and fractional-factorial designs

π ‚ ¬ ∫ “ ¬ · ≈ – · º π ¬ÿ∑ ∏ » “ µ √å «—§ ´’π ...nvi.ddc.moph.go.th/attach/ebook2.pdfπ ‚ ¬ ∫ “ ¬ · ≈ – · º π ¬ÿ∑ ∏ »

Π Π - w5. · PDF fileΠ Π Π Π 7kt5 501 7kt5 502 7kt5 504 7kt5 602 7kt5 604

2k Factorial Factorial Designsjain/cse567-08/ftp/k_172kd.pdf · 2k Factorial Factorial Designs Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 ... General 2k Factorial

Untitled-2 []”π” ”π— ß“π ≥– √√¡ “√ “√»÷ …“¢—Èπæ Èπ∞“π¡’Àπâ“∑’Ë √—∫º ‘¥ Õ∫„π “√‡∑’¬∫«ÿ

Ë °“√≈∫®”π«π Õß®”π«π∑’Ë¡’ μ—«μ —È߉¡ à‡° ‘π 9 · 76 ∫∑∑’Ë 4 °“√≈∫®”π«π Õß®”π«π∑’Ë¡’μ—«μ—È߉¡à‡°‘π

·π«∑“߇«™ªØ‘∫—μ‘„π°“√« ‘π‘®© · ·π«∑“߇«™ªØ‘∫—μ‘„π°“√« ‘π‘®© —¬·≈–√ —°…“ : ºŸâªÉ«¬

π“‚π‡∑§‚π‚≈¬’„π∏√√¡™“μ · °√°Æ“§¡ - °—𬓬π 2550 60 M T E C ¿“梬“¬· ¥ß¢π¢π“¥π“‚π (nanoscopic hairs)

À∑∏∆∏™∏ ™Ã∂∆π∫∞ ª∂ ∆√ ∫√π¡ø¡π∫√ ºÀ§√ ∫∞π ∆∏¡ ∫√π¡ø¡π∫∏ … · keºa§aio 3 1. ∂ÈÛ·ÁˆÁ‹ ™ËÌ›Ô

À°π∂π¡∏™ °π∞ ∆π™ ∂¶πÃ∂πƒ∏™∂π™ ª∞∑π∫∏™ ∂™∆π∞™∏™ … · Ο «Οδηγός Υγιεινής» εγκρίθηκε από

Introducción Modelo factorial ortogonal Construcción del modelo factorial: método de componentes principales Construcción del modelo factorial:

Título: ANÁLISIS FACTORIAL Presentación: El Análisis Factorial es

Measurements of K ± π + π - e ± ν ( Ke4) and K ± π ± π 0 π 0 decays at NA48/2

—ææ∑“π—ß ∏—¡¡∑“π—ß ™‘𓵑 · —ææ∑“π—ß ∏—¡¡∑“π—ß ™‘𓵑 °“√„Àâ∏√√¡–‡ªìπ∑“π ¬àÕ¡™π–°“√„Àâ∑—Èߪ«ß

OPDC 24 1 - home.kku.ac.th · PDF file°.æ.√. ”π—°ß“π§≥–°√√¡°“√æ —≤π“√–∫∫√“™°“√ §”π” π—∫µ—Èß·µàæ√–√“™°ƒÆ

Honors Pre-Cal Mrs. Cortez. xy = sin(x) 00 π /6½ π /21 5 π /6½ π 0 7 π /6-½-½ 3 π /2 11 π /6-½-½ 2π2π 0

a ( ) f x = + + a kx b kx k k 2 ∑ ∑ ∑ ( ) - in.tum.de · k n kj k f x n n kj k f x n n f x k j n f x kx f k f k f x kx n a f x kx dx π π π π π π π π π π π π π π

FACTORIAL AND FRACTIONAL FACTORIAL DESIGNS …pranjan/research/PHD_thesis.pdf · Abstract Two-level factorial and fractional factorial designs have played a prominent role in the

Factorial Designs Outlines: 2 2 factorial design 2 k factorial design, k>=3 Blocking and confounding in 2 k factorial design

CON 10 A · ·π«∑“ß°“√¥”‡π‘πß“π√–∫∫°“√¥ Ÿ·≈™ ૬‡À≈ Õπ —°‡√ ’¬π„π ∂“π» ÷°…“ æ‘¡æ å

[π-MACIERZATOR] [π-MACIERZATOR]

ª∂∆∞§§∞ ∫∞∆∂ƒ°∞™π∞ – ™À°∫√§§∏™∏ · 3 ¢π√π∫∏∆π∫√ ™Àªµ√À§π√ ∂§.π¡.À.∞.∂. ¶Úfi‰ÚÔ˜: ñ µ·Û›ÏÂÈÔ˜

FACTORIAL DESIGNS Two Factor Factorial Designs · 4 FACTORIAL DESIGNS 4.1 Two Factor Factorial Designs A two-factor factorial design is an experimental design in which data is collected