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Mathematical function
In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by ψ 1 ( z ) = d 2 d z 2 ln
Trigamma_function
Meromorphic function
functions have a pole of order m + 1. The function ψ(1)(z) is sometimes called the trigamma function. When m > 0 and Re z > 0, the polygamma function
Polygamma_function
Probability distribution
where the trigamma function, denoted ψ1(α), is the second of the polygamma functions, and is defined as the derivative of the digamma function: ψ 1 ( α
Beta_distribution
Topics referred to by the same term
{\displaystyle \psi (z)} the trigamma function ψ 1 ( z ) {\displaystyle \psi ^{1}(z)} and in physics to the quantum mechanical wave function. This disambiguation
Psi_function
Number, approximately 0.916
needed] G appears in values of the second polygamma function, also called the trigamma function, at fractional arguments: ψ 1 ( 1 4 ) = π 2 + 8 G ψ 1
Catalan's_constant
Inverse of the gamma function
{\displaystyle \psi ^{\left(1\right)}\left(x\right)} is the trigamma function. The inverse gamma function also has the following asymptotic formula Γ − 1 ( x
Inverse_gamma_function
Mathematical function
digamma function naturally appears when the derivative of the log-likelihood is taken for finding the maxima. Polygamma function Trigamma function Chebyshev
Digamma_function
Probability distribution
\psi ^{(1)}} is the trigamma function. This can be derived using the exponential family formula for the moment generating function of the sufficient statistic
Gamma_distribution
Family of probability distributions often used to model tails or extreme values
{\displaystyle \xi } only through the polygamma function of order 1 (also called the trigamma function): Var [ Y ] = { ψ ′ ( 1 ) − ψ ′ ( − 1 / ξ + 1
Generalized Pareto distribution
Generalized_Pareto_distribution
Rational number sequence
}B_{m}^{+}z^{m}} is an asymptotic series. It contains the trigamma function ψ1. From the generating functions above, one can obtain the following integral formula
Bernoulli_number
Generalization of gamma distribution to multiple dimensions
_{1}\left({\frac {n+1-i}{2}}\right)} where ψ 1 {\displaystyle \psi _{1}} is the trigamma function. This comes up when computing the Fisher information of the Wishart
Wishart_distribution
Summation formula
{1}{30n^{5}}}-{\frac {1}{42n^{7}}}+\cdots .} This is related to the trigamma function: ∑ k = 1 n 1 k 2 = π 2 6 − ψ ( 1 ) ( n + 1 ) {\displaystyle \sum _{k=1}^{n}{\frac
Euler–Maclaurin_formula
Probability distribution
where ψ {\displaystyle \psi } is the digamma function, ψ ′ {\displaystyle \psi '} is the trigamma function, and δ i j {\displaystyle \delta _{ij}} is the
Dirichlet_distribution
numbers List of physical constants Particular values of the Riemann zeta function Physical constant Both i and −i are roots of this equation, though neither
List of mathematical constants
List_of_mathematical_constants
_{-4})} Another closed form expression may be given in terms of the trigamma function: V = 3 3 ( ψ 1 ( 1 / 3 ) 2 − π 2 3 ) {\displaystyle V={\frac {\sqrt
Gieseking_manifold
Probability distribution
{\displaystyle \psi } and ψ ′ {\displaystyle \psi '} are the digamma function and trigamma function respectively. Given a value for β {\displaystyle \textstyle
Generalized normal distribution
Generalized_normal_distribution
approximation Spouge, John L. (1994). "Computation of the Gamma, Digamma, and Trigamma Functions". SIAM Journal on Numerical Analysis. 31 (3): 931–000. doi:10.1137/0731050
Spouge's_approximation
Probability distribution
in terms of the digamma ψ {\displaystyle \psi } and trigamma ψ ′ {\displaystyle \psi '} functions: μ i ∗ = ψ ( α i ) − ψ ( α D ) , i = 1 , … , D − 1 {\displaystyle
Logit-normal_distribution
) {\displaystyle \digamma ^{[1]}(\nu )} are values of digamma and trigamma functions at ν {\displaystyle \nu } , respectively. Demirhan and Hamurkaroglu
Generalized multivariate log-gamma distribution
Generalized_multivariate_log-gamma_distribution
Name for several different families of probability distributions
the digamma function, while ψ ′ = ψ ( 1 ) {\displaystyle \psi '=\psi ^{(1)}} is its first derivative, also known as the trigamma function, or the first
Generalized logistic distribution
Generalized_logistic_distribution
Swedish mathematician and politician (1814–1886)
constant, ζ stands for the Riemann zeta-function, Ψ is the digamma function, and Ψ1 is the trigamma function; see respectively eq. (43), (47) and (48)
Carl_Johan_Malmsten
Scottish mathematician
ISSN 1464-3839. Pairman, Eleanor. (1919). Tables of the digamma and trigamma functions. Cambridge University Press. OCLC 681090033. Pairman, Eleanor; Pearson
Eleanor_Pairman
TRIGAMMA FUNCTION
TRIGAMMA FUNCTION
Girl/Female
Indian, Telugu
Vishnu Sahastram
Male
Egyptian
, Functionary of the Interior.
Male
Celtic
, great justiciary, or functionary.
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Night
Male
Egyptian
, a great functionary.
Biblical
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Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Male
Egyptian
, the son of the functionary Heknofre.
Girl/Female
Hindu, Indian, Marathi, Sanskrit
Woman; A Pearl
Male
Egyptian
, an Egyptian functionary.
Girl/Female
Tamil
Triyama | தà¯à®°à®¿à®¯à®®à®¾
Night
Triyama | தà¯à®°à®¿à®¯à®®à®¾
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, a high Egyptian functionary.
TRIGAMMA FUNCTION
TRIGAMMA FUNCTION
Girl/Female
Irish
Excellent.
Girl/Female
Arabic
Most of Very Pious; Upright
Girl/Female
Teutonic
Inspired intelligence. Famous bearer: Veleda was a 1st century AD Germanic prophetess.
Boy/Male
American, Australian, British, Chinese, Danish, English, Finnish, German, Greek, Slovenia
Victory of the People; Abbreviation of Nicholas
Girl/Female
Arabic, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Muslim, Oriya, Sindhi, Telugu, Traditional
A Beautiful and Fragrant Flower; Fragrant
Surname or Lastname
English
English : variant of Sly.
Girl/Female
Tamil
Leafy
Girl/Female
Assamese, Danish, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Daffodil
Girl/Female
Indian
Having no enemy
Girl/Female
Hindu
TRIGAMMA FUNCTION
TRIGAMMA FUNCTION
TRIGAMMA FUNCTION
TRIGAMMA FUNCTION
TRIGAMMA FUNCTION
n.
The doctrine that all the functions of a living organism are due to an unknown vital principle distinct from all chemical and physical forces.
n.
The act of marrying, or the state of being married, three times; also, the offense of having three husbands or three wives at the same time.
n.
A syllogism with three conditional propositions, the major premises of which are disjunctively affirmed in the minor. See Dilemma.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.
pl.
of Functionary
a.
Pertaining to the function of an organ or part, or to the functions in general.
v. i.
To execute or perform a function; to transact one's regular or appointed business.
a.
Destitute of function, or of an appropriate organ. Darwin.
adv.
In a functional manner; as regards normal or appropriate activity.
a.
Having the digamma or its representative letter or sound; as, the Latin word vis is a digammated form of the Greek /.
v. i.
Alt. of Functionate
n.
A state of things in which it is difficult to determine which one of three courses to pursue.
a.
Pertaining to, or connected with, a function or duty; official.
n.
A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x2, 3x, Log. x, and Sin. x, are all functions of x.
v. t.
To assign to some function or office.
a.
Belonging or relating to life, either animal or vegetable; as, vital energies; vital functions; vital actions.
n.
Same as Eisel. F () F is the sixth letter of the English alphabet, and a nonvocal consonant. Its form and sound are from the Latin. The Latin borrowed the form from the Greek digamma /, which probably had the value of English w consonant. The form and value of Greek letter came from the Phoenician, the ultimate source being probably Egyptian. Etymologically f is most closely related to p, k, v, and b; as in E. five, Gr. pe`nte; E. wolf, L. lupus, Gr. ly`kos; E. fox, vixen ; fragile, break; fruit, brook, v. t.; E. bear, L. ferre. See Guide to Pronunciation, // 178, 179, 188, 198, 230.
n.
The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
n.
A Burman measure of twelve miles. V () V, the twenty-second letter of the English alphabet, is a vocal consonant. V and U are only varieties of the same character, U being the cursive form, while V is better adapted for engraving, as in stone. The two letters were formerly used indiscriminately, and till a comparatively recent date words containing them were often classed together in dictionaries and other books of reference (see U). The letter V is from the Latin alphabet, where it was used both as a consonant (about like English w) and as a vowel. The Latin derives it from it from a form (V) of the Greek vowel / (see Y), this Greek letter being either from the same Semitic letter as the digamma F (see F), or else added by the Greeks to the alphabet which they took from the Semitic. Etymologically v is most nearly related to u, w, f, b, p; as in vine, wine; avoirdupois, habit, have; safe, save; trover, troubadour, trope. See U, F, etc.
n.
A letter (/, /) of the Greek alphabet, which early fell into disuse.