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Senin, 10 Desember 2012

natural logaritma


Natural Logarithm

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The natural logarithm lnx is the logarithm having base e, where
 e=2.718281828....
(1)
This function can be defined
 lnx=int_1^x(dt)/t
(2)
for x>0.
NaturalLogEPlot
This definition means that e is the unique number with the property that the area of the region bounded by thehyperbola y=1/x, the x-axis, and the vertical lines x=1 and x=e is 1. In other words,
 int_1^e(dx)/x=lne=1.
(3)
The notation lnx is used in physics and engineering to denote the natural logarithm, while mathematicians commonly use the notation logx. In this work, lnx=log_ex denotes a natural logarithm, whereas logx=log_(10)xdenotes the common logarithm.
There are a number of notational conventions in common use for indication of a power of a natural logarithm. While some authors use ln^nz (i.e., using a trigonometric function-like convention), it is also common to write (lnz)^n.
Common and natural logarithms can be expressed in terms of each other as
lnx=(log_(10)x)/(log_(10)e)
(4)
log_(10)x=(lnx)/(ln10).
(5)
The natural logarithm is especially useful in calculus because its derivative is given by the simple equation
 d/(dx)lnx=1/x,
(6)
whereas logarithms in other bases have the more complicated derivative
 d/(dx)log_bx=1/(xlnb).
(7)
NaturalLogBranchCut
The natural logarithm can be analytically continued to complex numbers as
 lnz=ln|z|+iarg(z),
(8)
where |z| is the complex modulus and arg(z) is the complex argument. The natural logarithm is a multivalued function and hence requires a branch cut in the complex plane, which Mathematica's convention places at (-infty,0].
NaturalLogReImAbs
MinMax
Re
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The principal value of the natural logarithm is implemented in Mathematica as Log[x], which is equivalent to Log[E,x]. This function is illustrated above in the complex plane.
Note that the inverse trigonometric and inverse hyperbolic functions can be expressed (and, in fact, are commonlydefined) in terms of the natural logarithm, as summarized in the table below. Therefore, once these definition are agreed upon, the branch cut structure adopted for the natural logarithm fixes the branch cuts of these functions.
functionsymboldefinition
inverse cosecantcsc^(-1)z-iln(sqrt(1-1/(z^2)+i/z))
inverse cosinecos^(-1)z1/2pi+iln(iz+sqrt(1-z^2))
inverse cotangentcot^(-1)z1/2i[ln((z-i)/z)-ln((z+i)/z)]
inverse hyperbolic cosecantcsch^(-1)zln(sqrt(1+1/(z^2))+1/z)
inverse hyperbolic cosinecosh^(-1)zln(z+sqrt(z-1)sqrt(z+1))
inverse hyperbolic cotangentcoth^(-1)z1/2[ln(1+1/z)-ln(1-1/z)]
inverse hyperbolic secantsech^(-1)zln(sqrt(1/z-1)sqrt(1/z+1)+1/z)
inverse hyperbolic sinesinh^(-1)zln(z+sqrt(z^2+1))
inverse hyperbolic tangenttanh^(-1)z1/2[ln(1+z)-ln(1-z)]
inverse secantsec^(-1)z1/2pi+iln(sqrt(1-1/(z^2))+i/z)
inverse sinesin^(-1)z-iln(iz+sqrt(1-z^2))
inverse tangenttan^(-1)z1/2i[ln(1-iz)-ln(1+iz)]
 ln(1+x)=x-1/2x^2+1/3x^3-...
(9)
gives a Taylor series for the natural logarithm.
Continued fraction representations of logarithmic functions include
 ln(1+x)=x/(1+(1^2x)/(2+(1^2x)/(3+(2^2x)/(4+(2^2x)/(5+(3^2x)/(6+(3^2x)/(7+...)))))))
(10)
(Lambert 1770; Lagrange 1776; Olds 1963, p. 138; Wall 1948, p. 342) and
 ln((1+x)/(1-x))=(2x)/(1-(x^2)/(3-(4x^2)/(5-(9x^2)/(7-(16x^2)/(9-...)))))
(11)
(Euler 1813-1814; Wall 1948, p. 343; Olds 1963, p. 139).
For a complex number z, the natural logarithm satisfies
lnz=ln[re^(i(theta+2npi))]
(12)
=lnr+i(theta+2npi)
(13)
and
 PV(lnz)=lnr+itheta,
(14)
where PV is the principal value.
Some special values of the natural logarithm include
ln(-1)=ipi
(15)
ln0=-infty
(16)
ln1=0
(17)
lne=1
(18)
ln(+/-i)=+/-1/2pii.
(19)
Natural logarithms can sometimes be written as a sum or difference of "simpler" logarithms, for example
 ln(2+sqrt(3))=2ln(1+sqrt(3))-ln2,
(20)
which follows immediately from the identity
 2+sqrt(3)=1/2(1+sqrt(3))^2.
(21)
Plouffe (2006) found the following beautiful identities:
ln2=10sum_(n=1)^(infty)1/(n(e^(pin)+1))+6sum_(n=1)^(infty)1/(n(e^(pin)-1))-4sum_(n=1)^(infty)1/(n(e^(2pin)-1))
(22)
ln3=9sum_(n=1)^(infty)1/(n(e^(pin)-1))+(49)/3sum_(n=1)^(infty)1/(n(e^(pin)+1))-(14)/3sum_(n=1)^(infty)1/(n(e^(2pin)+1))+sum_(n=1)^(infty)1/(n(e^(3pin)-1))-7/3sum_(n=1)^(infty)1/(n(e^(3pin)+1))+2/3sum_(n=1)^(infty)1/(n(e^(6pin)+1))
(23)
ln5=(57)/4sum_(n=1)^(infty)1/(n(e^(pin)-1))+(91)/4sum_(n=1)^(infty)1/(n(e^(pin)+1))-(13)/2sum_(n=1)^(infty)1/(n(e^(2pin)+1))+3/4sum_(n=1)^(infty)1/(n(e^(5pin)-1))-7/4sum_(n=1)^(infty)1/(n(e^(5pin)+1))+1/2sum_(n=1)^(infty)1/(n(e^(10pin)+1)).
(24)

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