Greek letters used in mathematics, science, and engineering

Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. In these contexts, the capital letters and the small letters represent distinct and unrelated entities. Those Greek letters which have the same form as Latin letters are usually not used: capital A, B, E, H, I, K, M, N, O, P, T, X, Y, Z. Small ι (iota), ο (omicron) and υ (upsilon) are also rarely used, since they closely resemble the Latin letters i, o and u. Sometimes font variants of Greek letters are used as distinct symbols in mathematics, in particular for ε/ϵ (epsilon) and π/ϖ (pi). Additionally, despite being obsolete in Modern Greek, the archaic letter digamma (Ϝ/ϝ/ϛ) is sometimes used.

The Bayer designation naming scheme for stars typically uses the first Greek letter, α, for the brightest star in each constellation, and runs through the alphabet before switching to Latin letters.

In mathematical finance, the Greeks are the variables denoted by Greek letters used to describe the risk of certain investments.

The majority of English-speaking mathematicians use neither the modern nor the historical Greek pronunciations of the names of the letters, but the traditional English pronunciation, e.g. /ˈθeɪtə/ for θ, cf. ancient [tʰɛ̂ːta] and modern [ˈθita].

Typography

The Greek letter forms used in mathematics are often different from those used in Greek-language text: they are designed to be used in isolation, not connected to other letters, and some use variant forms which are not normally used in current Greek typography.

The OpenType font format has the feature tag 'mgrk' "Mathematical Greek" to identify a glyph as representing a Greek letter to be used in mathematical (as opposed to Greek language) contexts.

The table below shows a comparison of Greek letters rendered in TeX and HTML. The font used in the TeX rendering is an italic style. This is in line with the convention that variables should be italicized. As Greek letters are more often than not used as variables in mathematical formulas, a Greek letter appearing similar to the TeX rendering is more likely to be encountered in works involving mathematics.

Greek letters
Name	TeX	HTML		Name	TeX	HTML		Name	TeX	HTML		Name	TeX	HTML		Name	TeX	HTML
Alpha	$\Alpha \, \alpha \,$	Α α		Digamma	$\digamma \,$	Ϝ ϝ		Kappa	$\Kappa \, \kappa \, \varkappa \,$	Κ κ ϰ		Omicron	$\Omicron \, \omicron \,$	Ο ο		Upsilon	$\Upsilon \, \upsilon \,$	Υ υ
Beta	$\Beta \, \beta \,$	Β β		Zeta	$\Zeta \, \zeta \,$	Ζ ζ		Lambda	$\Lambda \, \lambda \,$	Λ λ		Pi	$\Pi \, \pi \, \varpi \,$	Π π ϖ		Phi	$\Phi \, \phi \, \varphi \,$	Φ ϕ φ
Gamma	$\Gamma \, \gamma \,$	Γ γ		Eta	$\Eta \, \eta \,$	Η η		Mu	$\Mu \, \mu \,$	Μ μ		Rho	$\Rho \, \rho \, \varrho \,$	Ρ ρ ϱ		Chi	$\Chi \, \chi \,$	Χ χ
Delta	$\Delta \, \delta \,$	Δ δ		Theta	$\Theta \, \theta \, \vartheta \,$	Θ θ ϑ		Nu	$\Nu \, \nu \,$	Ν ν		Sigma	$\Sigma \, \sigma \, \varsigma \,$	Σ σ ς		Psi	$\Psi \, \psi \,$	Ψ ψ
Epsilon	$\Epsilon \, \epsilon \, \varepsilon \,$	Ε ϵ ε		Iota	$\Iota \, \iota \,$	Ι ι		Xi	$\Xi \, \xi \,$	Ξ ξ		Tau	$\Tau \, \tau \,$	Τ τ		Omega	$\Omega \, \omega \,$	Ω ω

Concepts represented by a Greek letter

Αα (alpha)

• $\alpha$ represents:
  • the first angle in a triangle, opposite the side A
  • one root of a quadratic equation, where β represents the other
  • the ratio of collector current to emitter current in a bipolar junction transistor (BJT) in electronics
  • the statistical significance of a result
  • the false positive rate in statistics ("Type I" error)
  • the reciprocal of the sacrifice ratio
  • the fine structure constant in physics
  • the angle of attack of an aircraft
  • an alpha particle (He²⁺)
  • angular acceleration in physics
  • the linear thermal expansion coefficient
  • the thermal diffusivity
  • In organic chemistry the α-carbon is the backbone carbon next to the carbonyl carbon, most often for amino acids
  • right ascension in astronomy
  • the brightest star in a constellation
  • Iron Ferrite and numerous phases within materials science
  • the return in excess of the compensation for the risk borne in investment
  • the α-conversion in lambda calculus
  • the independence number of a graph

Ββ (beta)

• Β represents the beta function
• β represents:
  • the thermodynamic beta, equal to (k_BT)⁻¹, where k_B is Boltzmann's constant and T is the absolute temperature.
  • the second angle in a triangle, opposite the side B
  • one root of a quadratic equation, where α represents the other
  • the standardized regression coefficient for predictor or independent variables in linear regression (unstandardized regression coefficients are represented with the lower-case Latin b, but are often called "betas" as well)
  • the ratio of collector current to base current in a bipolar junction transistor (BJT) in electronics (current gain)
  • the false negative rate in statistics ("Type II" error)
  • the beta coefficient, the non-diversifiable risk, of an asset in mathematical finance
  • the sideslip angle of an airplane
  • the first-order effects of variations in Coriolis force with latitude in planetary dynamics
  • a beta particle (e⁻)
  • sound intensity
  • velocity divided by the speed of light in special relativity
  • the beta brain wave in brain or cognitive sciences
  • ecliptic latitude in astronomy
  • The ratio of plasma pressure to magnetic pressure in plasma physics
  • β-reduction in lambda calculus
  • The ratio of the velocity of an object to the speed of light as used in the Lorentz Factor
  • In organic chemistry, β represents the second carbon from a functional group

Γγ (gamma)

• Γ represents:
  • the circulation in fluid dynamics
  • the reflection coefficient of a transmission or telecommunication line.
  • the confinement factor of an optical mode in a waveguide
  • the gamma function, a generalization of the factorial
  • the upper incomplete gamma function
  • the modular group, the group of fractional linear transformations
  • the gamma distribution, a continuous probability distribution defined using the gamma function
  • second-order sensitivity to price in mathematical finance
  • the Christoffel symbols of the second kind
  • the neighbourhood of a vertex in a graph
  • the stack alphabet in the formal definition of a pushdown automaton
• γ represents:
  • the circulation strength in fluid dynamics
  • the partial safety factors applied to loads and materials in structural engineering
  • the specific weight of substances
  • the lower incomplete gamma function
  • the third angle in a triangle, opposite the side C
  • the Euler–Mascheroni constant in mathematics
  • gamma rays and the photon
  • the heat capacity ratio in thermodynamics
  • the Lorentz factor in special relativity
  • the damping constant (kg/s)

Δδ (delta)

• Δ represents:
  • a finite difference
  • a difference operator
  • a symmetric difference
  • the Laplace operator
  • the angle that subtends the arc of a circular curve in surveying
  • the determinant of an inverse matrix^[1]
  • the maximum degree of any vertex in a given graph
  • the difference or change in a given variable, e.g. ∆v means a difference or change in velocity
  • sensitivity to price in mathematical finance
  • distance to Earth, measured in astronomical units
  • heat in a chemical formula
  • the discriminant in the quadratic formula which determines the nature of the roots
  • the degrees of freedom in a non-pooled statistical hypothesis test of two population means
• δ represents:
  • percent error
  • a variation in the calculus of variations
  • the Kronecker delta function
  • the Feigenbaum constant
  • the force of interest in mathematical finance
  • the Dirac delta function
  • the receptor which enkephalins have the highest affinity for in pharmacology ^[2]
  • the Skorokhod integral in Malliavin calculus, a subfield of stochastic analysis
  • the minimum degree of any vertex in a given graph
  • a partial charge. δ− represents a negative partial charge, and δ+ represents a positive partial charge chemistry (See also: Solvation)
  • the Chemical shift of an atomic nucleus in NMR spectroscopy. For protons, this is relative to tetramethylsilane = 0.
  • stable isotope compositions
  • declination in astronomy
  • the Turner function in computational material science
  • depreciation in macroeconomics
  • noncentrality measure in statistics^[3]

Εε (epsilon)

• ε represents:
  • a small positive quantity; see limit
  • a random error in regression analysis
  • the absolute value of an error ^[4]
  • in set theory, the limit ordinal of the sequence $\omega,\omega^{\omega},\omega^{\omega^{\omega}},\dots$
  • in computer science, the empty string
  • the Levi-Civita symbol
  • in electromagnetics, dielectric permittivity
  • emissivity
  • strain in continuum mechanics
  • permittivity
  • the Earth's axial tilt in astronomy
  • elasticity in economics
  • expected value in probability theory and statistics
  • electromotive force
  • in chemistry, the molar extinction coefficient of a chromophore.
• set membership symbol ∈ is based on ε

Ϝϝ (digamma)

• Ϝ is sometimes used to represent the digamma function, though the Latin letter F (which is nearly identical) is usually substituted.

Ζζ (zeta)

• ζ represents:
  • the Riemann zeta function and other zeta functions in mathematics
  • the coefficient of viscous friction in polymer dynamics
  • the damping ratio
  • relative vertical vorticity in fluid dynamics

Ηη (eta)

• Η represents:
  • the Eta function of Ludwig Boltzmann's H-theorem ("Eta" theorem), in statistical mechanics
  • Information Theoretic (Shannon) Entropy
• η represents:
  • the intrinsic wave impedance of a medium (e.g. the impedance of free space)
  • the partial regression coefficient in statistics
  • elasticities in economics
  • the absolute vertical vorticity (relative vertical vorticity + Coriolis effect) in fluid dynamics
  • an index of refraction
  • the eta meson
  • viscosity
  • energy conversion efficiency
  • efficiency (physics)
  • the Minkowski metric tensor in relativity
  • noise in communication system models
  • η-conversion in lambda calculus
  • Cost-push supply side shocks in the Phillips Curve Equation (economics)
  • A right angle, i.e., π/2, as a follow-up to the tau/pi argument ^[5]

Θθ (theta)

• Θ (uppercase) represents:
  • an asymptotically tight bound related to big O notation.
  • Debye temperature in solid state physics
  • sensitivity to the passage of time in mathematical finance
  • in set theory, a certain ordinal number
  • in Econometrics and statistics, a space of parameters from which estimates are drawn
• θ (lowercase) represents:
  • a plane angle in geometry
  • the angle to the x axis in the xy-plane in spherical or cylindrical coordinates (mathematics)
  • the angle to the z axis in spherical coordinates (physics)
  • Bragg's angle of diffraction
  • the potential temperature in thermodynamics
  • the mean time between failure in reliability engineering
  • soil water contents in soil science
  • in mathematical statistics, an unknown parameter
  • theta functions
  • the angle of a scattered photon during a Compton scattering interaction
• ϑ ("script theta"), the cursive form of theta, often used in handwriting, represents
  • the first Chebyshev function in number theory

Ιι (iota)

• ι represents:
  • the index generator function in APL (in the form ⍳)
  • the orbital inclination in Celestial mechanics.

Κκ (kappa)

• Κ represents:
  • the Kappa number, indicating lignin content in pulp
• κ represents:
  • the Von Kármán constant, describing the velocity profile of turbulent flow
  • the kappa curve, a two-dimensional algebraic curve
  • the condition number of a matrix in numerical analysis
  • the connectivity of a graph in graph theory
  • curvature
  • dielectric constant $(\varepsilon / \varepsilon_0)$
  • thermal conductivity (usually a lowercase Latin k)
  • thermal diffusivity
  • a spring constant (usually a lowercase Latin k)
  • the heat capacity ratio in thermodynamics (usually γ)
  • the receptor which dynorphins have the highest affinity for in pharmacology^[2]

Λλ (lambda)

• Λ represents:
  • the von Mangoldt function in number theory
  • the set of logical axioms in the axiomatic method of logical deduction in first-order logic
  • the cosmological constant
  • the lambda baryon
  • a diagonal matrix of eigenvalues in linear algebra
  • the permeance of a material in electromagnetism
  • a lattice
• λ represents:
  • one wavelength of electromagnetic radiation
  • the decay constant in radioactivity
  • function expressions in the lambda calculus
  • a general eigenvalue in linear algebra
  • the expected number of occurrences in a Poisson distribution in probability
  • the arrival rate in queueing theory
  • the average lifetime or rate parameter in an exponential distribution (commonly used across statistics, physics, and engineering)
  • the failure rate in reliability engineering
  • the fundamental length of a fabrication process in VLSI design
  • the mean or average value (probability and statistics)
  • the latent heat of fusion
  • the lagrange multiplier in the mathematical optimization method, known as the shadow price in economics
  • the Lebesgue measure denotes the volume or measure of a Lebesgue measurable set
  • longitude in geodesy
  • linear density
  • ecliptic longitude in astronomy
  • the Liouville function in number theory
  • the Carmichael function in number theory
  • a unit of measure of volume equal to one microlitre (1 μL) or one cubic millimetre (1 mm³)
  • the empty string in formal grammar
  • binding of a variable in a function in Lambda calculus in computer science.

Μμ (mu)

• μ represents:
  • the Möbius function in number theory
  • the ring representation of a representation module
  • the population mean or expected value in probability and statistics
  • a measure in measure theory
  • micro-, an SI prefix denoting 10⁻⁶ (one millionth)
  • the coefficient of friction in physics
  • the service rate in queueing theory
  • the dynamic viscosity in physics
  • magnetic permeability in electromagnetics
  • a muon
  • reduced mass
  • chemical potential in condensed matter physics
  • the ion mobility in plasma physics
  • the Standard gravitational parameter in celestial mechanics

Νν (nu)

• ν represents:
  • frequency in physics in hertz (Hz)
  • Degrees of freedom in statistics
  • Poisson's ratio in material science
  • a neutrino
  • kinematic viscosity of liquids
  • stoichiometric coefficient in chemistry
  • dimension of nullspace in mathematics
  • True anomaly in Celestial Mechanics

Ξξ (xi)

• Ξ represents:
  • the original Riemann Xi function, i.e. Riemann's lower case ξ, as denoted by Edmund Landau and currently
  • the grand canonical ensemble found in statistical mechanics
  • the xi baryon
• ξ represents:
  • the original Riemann Xi function
  • the modified definition of Riemann xi function, as denoted by Edmund Landau and currently
  • a random variable
  • the extent of a chemical reaction
  • coherence length
  • the damping ratio
  • universal set

Οο (omicron)

• Ο represents:
  • big O notation (may be represented by an uppercase Latin O)
• o represents:
  • small o notation (may be represented by a lowercase Latin o)

Ππ (pi)

• Π represents:
  • the product operator in mathematics
  • a plane
  • the unary projection operation in relational algebra
  • osmotic pressure
• π represents:
  • Archimedes' constant, the ratio of a circle's circumference to its diameter
  • the prime-counting function
  • profit in microeconomics and game theory
  • inflation in macroeconomics, expressed as a constant with respect to time
  • the state distribution of a Markov chain
  • in reinforcement learning, a policy function defining how a software agent behaves for each possible state of its environment
  • a type of covalent bond in chemistry (pi bond)
  • a pion (pi meson) in particle physics
  • in statistics, the population proportion
  • nucleotide diversity in molecular genetics
  • in electronics, a special type of small signal model is referred to as a hybrid-pi model
  • in relational algebra for databases, represents projection
• ϖ (a graphic variant, see pomega) represents:
  • angular frequency of a wave, in fluid dynamics (angular frequency is usually represented by $\omega$ but this may be confused with vorticity in a fluid dynamics context)
  • longitude of pericenter, in astronomy^[6]
  • comoving distance, in cosmology^[7]

Ρρ (rho)

• Ρ represents:
  • one of the Gegenbauer functions in analytic number theory (may be replaced by the capital form of the Latin letter P).
• ρ represents:
  • one of the Gegenbauer functions in analytic number theory.
  • the Dickman-de Bruijn function
  • the radius in a polar, cylindrical, or spherical coordinate system
  • the correlation coefficient in statistics
  • the sensitivity to interest rate in mathematical finance
  • density (mass or charge per unit volume; may be replaced by the capital form of the Latin letter D)
  • resistivity
  • the shape and reshape operators in APL (in the form ⍴)
  • the utilization in queueing theory
  • the rank of a matrix

Σσ (sigma)

• Σ represents:
  • the summation operator
  • the covariance matrix
  • the set of terminal symbols in a formal grammar
• σ represents:
  • Stefan–Boltzmann constant in blackbody radiation
  • the divisor function in number theory
  • the real part of the complex variable s = σ + i t in analytic number theory
  • the sign of a permutation in the theory of finite groups
  • the population standard deviation, a measure of spread in probability and statistics
  • a type of covalent bond in chemistry (sigma bond)
  • the selection operator in relational algebra
  • stress in mechanics
  • electrical conductivity
  • area density
  • nuclear cross section
  • uncertainty
  • utilization in operations management
  • surface charge density for microparticles

Ττ (tau)

• τ (lower-case) represents:
  • torque, the rotational force in mechanics
  • the elementary tau lepton in particle physics
  • a mean lifetime, of an exponential decay or spontaneous emission process
  • the time constant of any device, such as an RC circuit
  • proper time in relativity
  • one turn: the constant ratio of a circle's circumference to its radius, with value 2π (6.283...).^[8]
  • Kendall tau rank correlation coefficient, a measure of rank correlation in statistics
  • Ramanujan's tau function in number theory
  • a measure of opacity, or how much sunlight cannot penetrate the atmosphere
  • the intertwining operator in representation theory
  • Shear stress in continuum mechanics
  • an internal system step in transition systems
  • a type variable in type theories, such as the simply typed lambda calculus
  • path tortuosity in reservoir engineering
  • in Topology, a given topology
  • the tau in biochemistry, a protein associated to microtubules
  • the golden ratio 1.618... (although φ (phi) is more common)
  • the number of divisors of highly composite numbers (sequence A000005 in OEIS)
  • in proton NMR spectroscopy, τ was formerly used for chemical shift

Υυ (upsilon)

• Υ represents:
  • the upsilon meson
• υ represents:
  • frequency in physics textbooks

Φφ (phi)

• Φ represents:
  • the work function in physics; the energy required by a photon to remove an electron from the surface of a metal
  • magnetic flux
  • the cumulative distribution function of the normal distribution in statistics
  • phenyl functional group
  • the reciprocal of the golden ratio (represented by φ, below), also represented as 1/φ
  • the value of the integration of information in a system (based on Integrated Information Theory)
  • note: a symbol for the empty set, $\varnothing$, resembles Φ but is not Φ
• φ represents:
  • the golden ratio 1.618... in mathematics, art, and architecture
  • Euler's totient function in number theory
  • a holomorphic map on an analytic space
  • the argument of a complex number in mathematics
  • the value of a plane angle in physics and mathematics
  • the angle to the z axis in spherical coordinates (mathematics)
  • the angle to the x axis in the xy-plane in spherical or cylindrical coordinates (physics)
  • latitude in geodesy
  • a scalar field
  • radiant flux
  • electric potential
  • the probability density function of the normal distribution in statistics
  • a feature of a syntactic node giving that node characteristics such as gender, number and person in syntax
  • the diameter of a vessel (engineering)
  • capacity reduction factor of materials in structural engineering

Χχ (chi)

• χ represents:
  • the chi distribution in statistics ($\chi^2$ is the more frequently encountered chi-squared distribution)
  • the chromatic number of a graph in graph theory
  • the Euler characteristic in algebraic topology
  • electronegativity in the periodic table
  • the Rabi frequency
  • the spinor of a fundamental particle
  • the Fourier transform of a linear response function
  • a character in mathematics; especially a Dirichlet character in number theory
  • the Sigma vectors in the unscented transform used in the unscented Kalman filter
  • sometimes the mole fraction
  • a characteristic or indicator function in mathematics

Ψψ (psi)

• Ψ represents:
  • water potential
  • a quaternary combinator in combinatory logic
• ψ represents:
  • the wave function in the Schrödinger equation of quantum mechanics
  • the stream function in fluid dynamics
  • yaw angle in vehicle dynamics
  • the angle between the x-axis and the tangent to the curve in the intrinsic coordinates system
  • the reciprocal Fibonacci constant
  • the second Chebyshev function in number theory
  • the polygamma function in mathematics
  • load combination factor in structural engineering

Ωω (omega)

• Ω represents:
  • the SI unit measure of electrical resistance, the ohm
  • angular velocity / radian frequency (rev/min)
  • the Right Ascension of the Ascending Node (RAAN) or Longitude of the ascending node in astronomy and Orbital Mechanics
  • the rotation rate of an object, particularly a planet, in dynamics
  • the Omega constant 0.5671432904097838729999686622...
  • an asymptotic lower bound related to big O notation
  • in probability theory and statistical mechanics, the set of possible distinct system states
  • a solid angle
  • the Omega baryon
  • the arithmetic function counting a number's prime factors
  • the density parameter in cosmology
• ω represents:
  • angular velocity / radian frequency (rad/sec)
  • the argument of periapsis in astronomy and Orbital Mechanics
  • a complex cube root of unity — the other is ω² — (used to describe various ways of calculating the discrete Fourier transform)
  • the differentiability class (i.e. $C^\omega$) for functions that are infinitely differentiable because they are complex analytic
  • the first infinite ordinal
  • the omega meson
  • the set of natural numbers in set theory (although $\mathbb{N}$ or N is more common in other areas of mathematics)
  • an asymptotically dominant quantity related to big O notation
  • in probability theory, a possible outcome of an experiment
  • in economics, the total wealth of an agent in general equilibrium theory
  • vertical velocity in pressure-based coordinate systems (commonly used in atmospheric dynamics)
  • the arithmetic function counting a number's distinct prime factors
  • a differential form (esp. on an analytic space)
  • the symbol ϖ, a graphic variant of π, is sometimes construed as omega with a bar over it; see π
  • The last carbon atom of a chain of carbon atoms is sometimes called the ω (omega) position, reflecting that ω is the last letter of the Greek alphabet. This nomenclature can be useful in describing unsaturated fatty acids.

See also

• English pronunciation of Greek letters
• Greek alphabet
• Greeks (finance) (Greek letters used in mathematical finance, including several invented names similar to Greek letters)
• Latin letters used in mathematics
• List of letters used in mathematics and science
• List of mathematical symbols
• Mathematical Alphanumeric Symbols (a Unicode block)
• Typographical conventions in mathematical formulae

References

1. http://www.maths.abdn.ac.uk/~igc/tch/eg1006/notes/node119.html
2. Katzung & Trevor's Pharmacology Examination & Board Review (9th Edition.). Anthony J. Trevor, Bertram G. Katzung, Susan B. Masters ISBN 978-0-07-170155-6. B. Opioid Peptides + 268 pp.
3. Applied Linear Statistical Models (5th ed.). Michael H. Kutner, Christopher J. Nachtsheim, John Neter, & William Li. New York: McGraw-Hill, 2005. ISBN 0-07-310874-X. xxviii + 1396 pp.
4. Golub, Gene; Charles F. Van Loan (1996). Matrix Computations – Third Edition. Baltimore: The Johns Hopkins University Press. p. 53. ISBN 0-8018-5413-X.
5. Pi, Tau and Eta. David Butler, Making Your Own Sense (blog). https://blogs.adelaide.edu.au/maths-learning/2011/06/08/pi-tau-and-eta/
6. Pomega - from Eric Weisstein's World of Physics
7. Outline for Weeks 14&15, Astronomy 225 Spring 2008
8. "Tau Day - No, really, pi is wrong: The Tau Manifesto by Michael Hartl". 2010. Retrieved 2015-03-20.

External links

• A pronunciation guide with audio