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For positive integer m the derivative of gamma function can be calculated as follows ( here γ is the Euler – Mascheroni constant ):

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## Some Related Sentences

For and positive

__For__this change

**is**not a change from one

__positive__position to another, but a change from order and truth to disorder and negation.

__For__example, one study on volunteerism found that feeling overwhelmed by others ' demands had an even stronger negative effect on mental health than helping had a

__positive__one

**(**although

__positive__effects were still significant ).

Also, supporters

**of**this view would characterize Luke ’ s portrayal**of****the**Roman Empire**as**__positive__because they believe Luke “ glosses over negative aspects**of****the**empire and presents imperial power positively .”__For__example, when Paul**is**before**the**council defending himself, Paul says that he**is**“ on trial concerning**the**hope**of****the**resurrection**of****the**dead ”**(**Acts 23: 6 ).__For__many applications where one

**is**only concerned about rotation around one axis, it

**is**sufficient to discard

**the**pseudovector nature

**of**angular momentum, and treat it like a scalar where it

**is**

__positive__when it corresponds to a counter-clockwise rotation, and negative clockwise.

*

__For__a finite field**of**prime order p,**the**algebraic closure**is**a countably infinite field which contains a copy**of****the**field**of**order p < sup > n </ sup > for each__positive__**integer**n**(**and**is**in fact**the**union**of**these copies ).__For__

__positive__values

**of**a and b,

**the**binomial theorem with n = 2

**is**

**the**geometrically evident fact that a square

**of**side

**can**

**be**cut into a square

**of**side a, a square

**of**side b, and two rectangles with sides a and b. With n = 3,

**the**theorem states that a cube

**of**side

**can**

**be**cut into a cube

**of**side a, a cube

**of**side b, three a × a × b rectangular boxes, and three a × b × b rectangular boxes.

__For__example, if an electric field

**is**placed across a solution

**of**Na < sup >+</ sup > and Cl < sup >−</ sup >

**(**and conditions are right )

**the**sodium ions move towards

**the**negative electrode

**(**cathode ), while

**the**chloride ions move towards

**the**

__positive__electrode

**(**anode ).

__For__example, reversing

**the**current direction in a Daniell galvanic cell would produce an electrolytic cell, where

**the**copper electrode

**is**

**the**

__positive__terminal and

**the**anode.

__For__a given material, it

**can**have a

__positive__or negative sign or exceptionally it

**can**

**be**zero, and this

**can**depend on

**the**temperature,

**as**it does for water about 4 C. The concept

**of**latent heat with respect to volume was perhaps first recognized by Joseph Black in 1762.

__For__

**the**simple electric dipole given above,

**the**electric dipole moment points from

**the**negative charge towards

**the**

__positive__charge, and has a magnitude equal to

**the**strength

**of**each charge times

**the**separation between

**the**charges.

__For__larger values

**of**n, Fermat's Last Theorem states there are no

__positive__

**integer**solutions

**(**x, y, z ).</ td >

__For__example, air pollution may generate a negative externality, and education may generate a

__positive__externality

**(**less crime, etc .).

__For__any

__positive__

**integer**n,

**the**set

**of**all n-tuples

**of**real numbers forms an n-dimensional vector space over R, which

**is**denoted R < sup > n </ sup > and sometimes called real coordinate space.

__For__a reason that was not recorded, he identified

**the**term "

__positive__" with vitreous electricity and " negative " with resinous electricity.

__For__example, one

**of**

**the**most recent results

**is**that, even after subtracting

**the**

__positive__influence

**of**decadal variation, shown to

**be**possibly present in

**the**ENSO trend,

**the**amplitude

**of**

**the**ENSO variability in

**the**observed data still increases, by

**as**much

**as**60 % in

**the**last 50 years.

__For__closed

**(**orientable or non-orientable ) surfaces with

__positive__genus,

**the**maximum number p

**of**colors needed depends on

**the**surface's

**Euler**characteristic χ according to

**the**formula

__For__s-polarization, a

__positive__r or t means that

**the**electric fields

**of**

**the**incoming and reflected or transmitted wave are parallel, while negative means anti-parallel.

__For__p-polarization, a

__positive__r or t means that

**the**magnetic fields

**of**

**the**waves are parallel, while negative means anti-parallel.

*

__For__every prime number p and__positive__**integer**n, there exists a finite field with p < sup > n </ sup > elements.__For__any

__positive__integers and, there exists a graph with girth at least and chromatic number at least ; for instance,

**the**Grötzsch graph

**is**triangle-free and has chromatic number 4, and repeating

**the**Mycielskian construction used to form

**the**Grötzsch graph produces triangle-free graphs

**of**arbitrarily large chromatic number.

__For__complex numbers with a

__positive__real part, it

**is**defined via an improper integral that converges:

For and integer

__For__example, an array

**of**10

__integer__variables, with indices 0 through 9, may

**be**stored

**as**10 words at memory addresses 2000, 2004, 2008, … 2036, so that

**the**element with index i has

**the**address 2000 + 4 × i.

__For__example,

**the**division example above

**is**surjective

**(**or onto ) because every rational number may

**be**expressed

**as**a quotient

**of**an

__integer__and a natural number.

__For__example, algorithms are known for factoring an n-bit

__integer__using just over 2n qubits

**(**Shor's algorithm ).

__For__

__integer__order α = n, J < sub > n </ sub >

**is**often defined via a Laurent series for a generating

**function**:

__For__given nonzero integers a and b there

**is**a nonzero

__integer__

**of**minimal absolute value among all those

**of**

**the**form ax + by with x and y integers ; one

**can**assume d > 0 by changing

**the**signs

**of**both s and t if necessary.

__For__example, a 1039 bit

__integer__was factored with

**the**special number field sieve using 400 computers over 11 months.

Some sets are infinite ; these sets have more than n elements for any

__integer__n.__For__example,**the**set**of**natural numbers, denotable by, has infinitely many elements, and we cannot use any normal number to give its size.__For__example, intervals, where takes all

__integer__values in Z, cover R but there

**is**no finite subcover.

__For__instance,

**can**hold an unboxed

__integer__in a range supported by

**the**hardware and implementation, permitting more efficient arithmetic than on big integers or arbitrary precision types.

__For__that purpose, one needs a hash

**function**that maps similar keys to hash values that differ by at most

**m**, where

**m**

**is**a small

__integer__

**(**say, 1 or 2 ).

__For__example, when mapping character strings between upper and lower case, one

**can**use

**the**binary encoding

**of**each character, interpreted

**as**an

__integer__, to index a table that gives

**the**alternative form

**of**that character (" A " for " a ", " 8 " for " 8 ", etc .).

__For__instance, suppose that each input

**is**an

__integer__z in

**the**range 0 to N − 1, and

**the**output must

**be**an

__integer__h in

**the**range 0 to n − 1, where N

**is**much larger than n. Then

**the**hash

**function**could

**be**h

__For__shapes that are smooth, or shapes with a small number

**of**corners,

**the**shapes

**of**traditional geometry and science,

**the**Hausdorff dimension

**is**an

__integer__.

__For__instance, Fermat's little theorem for

**the**nonzero integers modulo a prime generalizes to Euler's theorem for

**the**invertible numbers modulo any nonzero

__integer__, which generalizes to Lagrange's theorem for finite groups.

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