Xi (letter)

Xi (uppercase Ξ, lowercase ξ; Greek: ξι) is the 14th letter of the Greek alphabet. It is pronounced [ksi] in Modern Greek, and generally /ˈzaɪ/ or /ˈsaɪ/ in English.^[1] In the system of Greek numerals, it has a value of 60. Xi was derived from the Phoenician letter samekh .

Xi is not to be confused with the letter chi, which gave its form to the Latin letter X.

Greek

A joined variant of Ξ.

Both in classical Ancient Greek and in Modern Greek, the letter Ξ represents the sound [ks]. In some archaic local variants of the Greek alphabet, this letter was missing. Instead, especially in the dialects of most of the Greek mainland and Euboea, the sound [ks] was represented by Χ (which in classical Greek is chi, used for /kʰ/). Because this variant of the Greek alphabet was used in Italy, the Latin alphabet borrowed Χ rather than Ξ as the Latin letter X.

Cyrillic

While having no Latin derivative, the Xi was adopted into the early Cyrillic alphabet, as the letter ksi (Ѯ, ѯ).

Mathematics and science

Upper case

The upper-case letter Ξ is used as symbol for:

• The "cascade particles" in particle physics
• Indicating "no change of state" in Z notation
• The partition function under the grand canonical ensemble in statistical mechanics
• Harish-Chandra's Ξ function
• The Riemann Xi function
• Ether currency of Ethereum

Lower case

The lower-case letter ξ is used as a symbol for:

• Random variables
• Propositional variables in some philosophical works, first found in Wittgenstein's Tractatus
• Dimensionless distance variable used in the Lane–Emden equation
• Extent of reaction, a concept in physical chemistry used most often in chemical engineering kinetics
• Eigenvectors
• Damping Ratio C/Ccr(vibrational analysis)
• A parameter in a generalized Pareto distribution
• The symmetric function equation of the Riemann zeta function in mathematics, also known as the Riemann Xi function
• A universal set in set theory
• A number used in the remainder term of Taylor's theorem that falls between the limits a and b
• A number used in error approximations for formulas that are applications of Taylor's theorem, such as Newton–Cotes formulas
• One of the two different polypeptide chains of the human embryonic hemoglobin types Hb-Portland (ξ₂γ₂) and Hb-Gower I (ξ₂ε₂)
• The correlation function in astronomy
• Spatial frequency;^[2] also sometimes temporal frequency.
• A small displacement in MHD plasma stability theory
• A parameter denoted as warped time used to derive the equations for homogeneous azeotropic distillation
• The x-coordinate of computational space as used in computational fluid dynamics
• Potential difference in physics (in volts)
• The radial integral in the spin-orbit matrix operator in atomic physics.
• The Killing vector in general relativity.
• Average logarithmic energy decrement per collision (neutron calculations in nuclear physics)
• Pippard's cohesion length in superconductors
• State Price Density in mathematical finance
• The diameter of a crystal nucleus in nucleation theory
• Microturbulence velocity in a stellar atmosphere

Character Encodings

• Greek Xi / Coptic Ksi

Character	Ξ		ξ		Ⲝ		ⲝ
Unicode name	GREEK CAPITAL LETTER XI		GREEK SMALL LETTER XI		COPTIC CAPITAL LETTER KSI		COPTIC SMALL LETTER KSI
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	926	U+039E	958	U+03BE	11420	U+2C9C	11421	U+2C9D
UTF-8	206 158	CE 9E	206 190	CE BE	226 178 156	E2 B2 9C	226 178 157	E2 B2 9D
Numeric character reference	Ξ	Ξ	ξ	ξ	Ⲝ	Ⲝ	ⲝ	ⲝ
Named character reference	Ξ		ξ
DOS Greek	141	8D	165	A5
DOS Greek-2	189	BD	232	E8
Windows 1253	206	CE	238	EE
TeX	\Xi		\xi

^[3]

• Mathematical Xi

Character	𝚵		𝛏		𝛯		𝜉		𝜩		𝝃
Unicode name	MATHEMATICAL BOLD CAPITAL XI		MATHEMATICAL BOLD SMALL XI		MATHEMATICAL ITALIC CAPITAL XI		MATHEMATICAL ITALIC SMALL XI		MATHEMATICAL BOLD ITALIC CAPITAL XI		MATHEMATICAL BOLD ITALIC SMALL XI
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	120501	U+1D6B5	120527	U+1D6CF	120559	U+1D6EF	120585	U+1D709	120617	U+1D729	120643	U+1D743
UTF-8	240 157 154 181	F0 9D 9A B5	240 157 155 143	F0 9D 9B 8F	240 157 155 175	F0 9D 9B AF	240 157 156 137	F0 9D 9C 89	240 157 156 169	F0 9D 9C A9	240 157 157 131	F0 9D 9D 83
UTF-16	55349 57013	D835 DEB5	55349 57039	D835 DECF	55349 57071	D835 DEEF	55349 57097	D835 DF09	55349 57129	D835 DF29	55349 57155	D835 DF43
Numeric character reference	𝚵	𝚵	𝛏	𝛏	𝛯	𝛯	𝜉	𝜉	𝜩	𝜩	𝝃	𝝃

Character	𝝣		𝝽		𝞝		𝞷
Unicode name	MATHEMATICAL SANS-SERIF BOLD CAPITAL XI		MATHEMATICAL SANS-SERIF BOLD SMALL XI		MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL XI		MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL XI
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	120675	U+1D763	120701	U+1D77D	120733	U+1D79D	120759	U+1D7B7
UTF-8	240 157 157 163	F0 9D 9D A3	240 157 157 189	F0 9D 9D BD	240 157 158 157	F0 9D 9E 9D	240 157 158 183	F0 9D 9E B7
UTF-16	55349 57187	D835 DF63	55349 57213	D835 DF7D	55349 57245	D835 DF9D	55349 57271	D835 DFB7
Numeric character reference	𝝣	𝝣	𝝽	𝝽	𝞝	𝞝	𝞷	𝞷

These characters are used only as mathematical symbols. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.

Other uses

Upper case Ξ is used as an 'E' to stylize company names/logos like Razer (styled as RΛZΞR) and musician Banners (styled as BANNΞRS)

References

1. "xi". New Oxford American Dictionary, 2nd Edition.
2. SPIE Optipedia article: "Spatial Frequency"
3. Unicode Code Charts: Greek and Coptic (Range: 0370-03FF)