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If ƒ is complex differentiable at every point z < sub > 0 </ sub > in an open set U, we say that ƒ is holomorphic on U. We say that ƒ is holomorphic at the point z < sub > 0 </ sub > if it is holomorphic on some neighborhood of z < sub > 0 </ sub >.

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## Some Related Sentences

If and ƒ

__If__

**is**continuous

**in**

**an**

**open**

**set**Ω and

**the**partial derivatives

**of**

__ƒ__with respect to x and y exist

**in**Ω

**,**and satisfies

**the**Cauchy – Riemann equations throughout Ω

**,**then

__ƒ__

**is**

**holomorphic**( and thus analytic ).

*

__If____ƒ__(**z**)**is**locally integrable**in****an****open**domain Ω ⊂ C**,**and satisfies**the**Cauchy – Riemann equations weakly**,**then__ƒ__agrees almost everywhere with**an**analytic function**in**Ω**.**__If__X

**is**a

**set**and M

**is**a complete metric space

**,**then

**the**

**set**B ( X

**,**M )

**of**all bounded functions

__ƒ__from X to M

**is**a complete metric space

**.**

__If__X

**is**a topological space and M

**is**a complete metric space

**,**then

**the**

**set**C

**<**

**sub**

**>**b

**</**

**sub**>( X

**,**M ) consisting

**of**all continuous bounded functions

__ƒ__from X to M

**is**a closed subspace

**of**B ( X

**,**M ) and hence also complete

**.**

In mathematics

**,****an**inverse function**is**a function**that**undoes another function:__If__**an**input x into**the**function__ƒ__produces**an**output y**,**then putting y into**the**inverse function g produces**the**output x**,**and vice versa**.**__If__

__ƒ__

**is**invertible

**,**

**the**function g

**is**unique ;

**in**other words

**,**there can be

**at**most one function g satisfying this property

**.**

__If__

**the**domain

**is**

**the**real numbers

**,**then each element

**in**Y would correspond to two different elements

**in**X (± x ), and therefore

__ƒ__would not be invertible

**.**

__If__

**an**inverse function exists for a given function

__ƒ__

**,**

**it**

**is**unique:

**it**must be

**the**inverse relation

**.**

__If__

__ƒ__: A → B

**is**a homomorphism between two algebraic structures ( such as homomorphism

**of**groups

**,**or a linear map between vector spaces ), then

**the**relation ≡ defined by

* Suppose

**that****is**a sequence**of**Lipschitz continuous mappings between two metric spaces**,**and**that**all have Lipschitz constant bounded by**some**K**.**__If____ƒ__**<****sub****>**n**</****sub****>**converges to a mapping__ƒ__uniformly**,**then__ƒ__**is**also Lipschitz**,**with Lipschitz constant bounded by**the**same K**.**In particular**,**this implies**that****the****set****of**real-valued functions**on**a compact metric space with a particular bound for**the**Lipschitz constant**is**a closed and convex subset**of****the**Banach space**of**continuous functions**.**

If and is

__If__his dancers are sometimes made to look as

**if**they might be creatures from Mars

**,**this

__is__consistent with his intention

**of**placing them

**in**

**the**orbit

**of**another world

**,**a world

**in**which they are freed

**of**their pedestrian identities

**.**

__If__a work

__is__divided into several large segments

**,**a last-minute drawing

**of**random numbers may determine

**the**order

**of**

**the**segments for any particular performance

**.**

__If__Wilhelm Reich

__is__

**the**Moses who has led them out

**of**

**the**Egypt

**of**sexual slavery

**,**Dylan Thomas

__is__

**the**poet who offers them

**the**Dionysian dialectic

**of**justification for their indulgence

**in**liquor

**,**marijuana

**,**sex

**,**and jazz

**.**

__If__he

__is__

**the**child

**of**nothingness

**,**

**if**he

__is__

**the**predestined victim

**of**

**an**age

**of**atomic wars

**,**then he will consult only his own organic needs and go beyond good and evil

**.**

__If__he thus achieves a lyrical

**,**dreamlike

**,**drugged intensity

**,**he pays

**the**price for his indulgence by producing work -- Allen Ginsberg's `` Howl ''

__is__a striking example

**of**this tendency --

**that**

__is__disoriented

**,**Dionysian but without depth and without Apollonian control

**.**

__If__love reflects

**the**nature

**of**man

**,**as Ortega Y Gasset believes

**,**

**if**

**the**person

**in**love betrays decisively what he

__is__by his behavior

**in**love

**,**then

**the**writers

**of**

**the**beat generation are creating a new literary genre

**.**

__If__

**the**existent form

__is__to be retained new factors

**that**reinforce

**it**must be introduced into

**the**situation

**.**

__If__

**we**remove ourselves for a moment from our time and our infatuation with mental disease

**,**isn't there something absurd about a hero

**in**a novel who

__is__defeated by his infantile neurosis??

__If__many

**of**

**the**characters

**in**contemporary novels appear to be

**the**bloodless relations

**of**characters

**in**a case history

**it**

__is__because

**the**novelist

__is__often forgetful today

**that**those things

**that**

**we**call character manifest themselves

**in**surface behavior

**,**

**that**

**the**ego

__is__still

**the**executive agency

**of**personality

**,**and

**that**all

**we**know

**of**personality must be discerned through

**the**ego

**.**

__If__our sincerity

__is__granted

**,**and

**it**

__is__granted

**,**

**the**discrepancy can only be explained by

**the**fact

**that**

**we**have come to believe hearsay and legend about ourselves

**in**preference to

**an**understanding gained by earnest self-examination

**.**

If and complex

__If__

**we**are discussing

**differentiable**complex-valued functions

**,**then Af and V are

__complex__vector spaces

**,**and Af may be any

__complex__numbers

**.**

__If__X

**is**a Banach space and K

**is**

**the**underlying field ( either

**the**real or

**the**

__complex__numbers ), then K

**is**itself a Banach space ( using

**the**absolute value as norm ) and

**we**can define

**the**continuous dual space as X ′ = B ( X

**,**K ),

**the**space

**of**continuous linear maps into K

**.**

__If__more than one eigenvalue

**is**negative

**,**then

**the**stationary

**point**

**is**a more

__complex__one

**,**and

**is**usually

**of**little interest

**.**

__If__a simply stable system response neither decays nor grows over time

**,**and has no oscillations

**,**

**it**

**is**marginally stable:

**in**this case

**the**system transfer function has non-repeated poles

**at**

__complex__plane origin ( i

**.**e

**.**their real and

__complex__component

**is**zero

**in**

**the**continuous time case ).

__If__

**the**weight

**of**

**the**cable and supporting wires are not negligible then

**the**analysis

**is**more

__complex__

**.**

__If__a stitch

**in**a finished crocheted item breaks

**,**

**the**stitches above and below remain intact

**,**and because

**of**

**the**

__complex__looping

**of**each stitch

**,**

**the**stitches

**on**either side are unlikely to come loose unless heavily stressed

**.**

__If__

**we**look for solutions

**that**have

**the**form

**,**where C

**is**a constant

**,**

**we**discover

**the**relationship

**,**and thus must be one

**of**

**the**

__complex__numbers or

**.**

__If__

**<**math

**>**c ^ 2

**<**4km

**</**math

**>**there are two

__complex__conjugate roots a ± ib

**,**and

**the**solution ( with

**the**above boundary conditions ) will look like this:

__If__anything

**,**

**the**research has shown

**the**situation

**is**even more

__complex__than initially imagined

**,**and

**we**believe not enough

**is**known to structure

**the**classification

**of**psychiatric disorders according to etiology

**.**

__If__

**the**obstructing object provides multiple

**,**closely spaced openings

**,**a

__complex__pattern

**of**varying intensity can result

**.**

__If__

**the**ether

**is**part

**of**a more

__complex__molecule

**,**

**it**

**is**described as

**an**alkoxy substituent

**,**so-OCH

**<**

**sub**

**>**3

**</**

**sub**

**>**would be considered a " methoxy -" group

**.**

__If__

**the**cell

**is**synthesizing DNA each pore

__complex__needs to transport about 100 histone molecules per minute

**.**

__If__

**the**cell

**is**growing rapidly

**,**each

__complex__also needs to transport about 6 newly assembled large and small ribosomal subunits per minute from

**the**nucleus to

**the**cytosol

**,**where they are used to synthesize proteins

**.**

__If__

**it**

**is**unaffected

**,**then

**it**

**is**a simple partial seizure ; otherwise

**it**

**is**a

__complex__partial ( psychomotor ) seizure

**.**

__If__a

__complex__function =

**is**

**holomorphic**

**,**then u and v have first partial derivatives with respect to x and y

**,**and satisfy

**the**Cauchy – Riemann equations:

In The Origin

**of**Species**,**he wrote**,**"__If__**it**could be demonstrated**that**any__complex__organ existed**,**which could not possibly have been formed by numerous**,**successive**,**slight modifications**,**my theory would absolutely break down**.**__If__( i ) or ( j ) have side effects or introduce run time errors

**,**

**the**subtle differences between ( l ) and ( l ') can be

__complex__

**.**

__If__

**,**however

**,**

**the**

**set**

**of**allowed candidates

**is**expanded to

**the**

__complex__numbers

**,**

**every**non-constant polynomial has

**at**least one root ; this

**is**

**the**fundamental theorem

**of**algebra

**.**

*

__If__R denotes**the**ring CY**of**polynomials**in**two variables with__complex__coefficients**,**then**the**ideal generated by**the**polynomial Y**<**sup**>**2**</**sup**>**− X**<**sup**>**3**</**sup**>**− X − 1**is**a prime ideal ( see elliptic curve ).0.190 seconds.