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erf

The Error Function

erfc

The Complementary Error Function and its Iterated Integrals

erfi

The Imaginary Error Function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

erf(x)

erfc(x)

erfc(n, x)

erfi(x)

Parameters

x

-

algebraic expression

n

-

algebraic expression, understood to be an integer  1

Description

• 

The error function is defined for all complex x by

erfx=20xⅇt2ⅆtπ

• 

The complementary error function is defined by

erfcx=1erfx=12π120xⅇt2ⅆt

• 

The iterated integrals of the complementary error function are defined by

erfc1,x=2πⅇx2

erfcn,x=xerfcn1,tⅆtn0

  

(Note erfc0,x=erfcx.)

• 

The imaginary error function is defined by

erfix=IerfIx=2π0xⅇt2ⅆt

• 

All of these functions are entire.

Examples

erf∞

1

(1)

erf3

erf3

(2)

evalf

0.9999779095

(3)

erfc3.

0.00002209049700

(4)

erf1.1.I

1.3161512820.1904534692I

(5)

erfc1.52.85I

62.8206488910.56167495I

(6)

ⅆⅆxerfx

2ⅇx2π

(7)

ⅆⅆxerfc5,x

erfc4,x

(8)

erfix

erfix

(9)

serieserfix,x,4

2πx+23πx3+Ox4

(10)

expanderfc2,x,x

12x212x2erfx12xⅇx2π+1414erfx

(11)

convert,erfc

12x212x21erfcx12xⅇx2π+14erfcx

(12)

References

  

Erdelyi, A. Higher Transcendental Functions. McGraw-Hill, 1953. Vol. 2.

See Also

convert

dawson

Fresnel

initialfunctions

 


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