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4 Elementary FunctionsLogarithm, Exponential, Powers

§4.2 Definitions

Contents

§4.2(i) The Logarithm

The general logarithm function Lnz is defined by

4.2.1 Lnz=1zdtt,
z0,

where the integration path does not intersect the origin. This is a multivalued function of z with branch point at z=0.

The principal value, or principal branch, is defined by

4.2.2 lnz=1zdtt,

where the path does not intersect (-,0]; see Figure 4.2.1. lnz is a single-valued analytic function on \(-,0] and real-valued when z ranges over the positive real numbers.

See accompanying text
Figure 4.2.1: z-plane: Branch cut for lnz and zα. Magnify

The real and imaginary parts of lnz are given by

4.2.3 lnz=ln|z|+iphz,
-π<phz<π.

For phz see §1.9(i).

The only zero of lnz is at z=1.

Most texts extend the definition of the principal value to include the branch cut

4.2.4 z=x,
-<x<0,

by replacing (4.2.3) with

4.2.5 lnz=ln|z|+iphz,
-π<phzπ.

With this definition the general logarithm is given by

4.2.6 Lnz=lnz+2kπi,

where k is the excess of the number of times the path in (4.2.1) crosses the negative real axis in the positive sense over the number of times in the negative sense.

In the DLMF we allow a further extension by regarding the cut as representing two sets of points, one set corresponding to the “upper side” and denoted by z=x+i0, the other set corresponding to the “lower side” and denoted by z=x-i0. Again see Figure 4.2.1. Then

4.2.7 ln(x±i0)=ln|x|±iπ,
-<x<0,

with either upper signs or lower signs taken throughout. Consequently lnz is two-valued on the cut, and discontinuous across the cut. We regard this as the closed definition of the principal value.

In contrast to (4.2.5) the closed definition is symmetric. As a consequence, it has the advantage of extending regions of validity of properties of principal values. For example, with the definition (4.2.5) the identity (4.8.7) is valid only when |phz|<π, but with the closed definition the identity (4.8.7) is valid when |phz|π. For another example see (4.2.37).

In the DLMF it is usually clear from the context which definition of principal value is being used. However, in the absence of any indication to the contrary it is assumed that the definition is the closed one. For other examples in this chapter see §§4.23, 4.24, 4.37, and 4.38.

§4.2(ii) Logarithms to a General Base a

With a,b0 or 1,

4.2.8 logaz =lnz/lna,
4.2.9 logaz =logbzlogba,
4.2.10 logab =1logba.

Natural logarithms have as base the unique positive number

4.2.11 e=2.71828 18284 59045 23536

such that

4.2.12 lne=1.

Equivalently,

4.2.13 1edtt=1.

Thus

4.2.14 logez=lnz,
4.2.15 log10z=(lnz)/(ln10)=(log10e)lnz,
4.2.16 lnz=(ln10)log10z,
4.2.17 log10e=0.43429 44819 03251 82765,
4.2.18 ln10=2.30258 50929 94045 68401.

logex=lnx is also called the Napierian or hyperbolic logarithm. log10x is the common or Briggs logarithm.

§4.2(iii) The Exponential Function

4.2.19 expz=1+z1!+z22!+z33!+.

The function exp is an entire function of z, with no real or complex zeros. It has period 2πi:

4.2.20 exp(z+2πi)=expz.

Also,

4.2.21 exp(-z)=1/exp(z).
4.2.22 |expz|=exp(z).

The general value of the phase is given by

4.2.23 ph(expz)=z+2kπ,
k.

If ζ0 then

4.2.25 expz=ζz=Lnζ.

§4.2(iv) Powers

Powers with General Bases

The general ath power of z is defined by

4.2.26 za=exp(aLnz),
z0.

In particular, z0=1, and if a=n=1,2,3,, then

4.2.27 za=zzzn times=1/z-a.

In all other cases, za is a multivalued function with branch point at z=0. The principal value is

4.2.28 za=exp(alnz).

This is an analytic function of z on \(-,0], and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless a.

4.2.29 |za|=|z|aexp(-(a)phz),
4.2.30 ph(za)=(a)phz+(a)ln|z|,

where phz[-π,π] for the principal value of za, and is unrestricted in the general case. When a is real

4.2.31 |za| =|z|a,
ph(za) =aphz.

Unless indicated otherwise, it is assumed throughout the DLMF that a power assumes its principal value. With this convention,

4.2.32 ez=expz,

but the general value of ez is

4.2.33 ez=(expz)exp(2kzπi),
k.

For z=1

4.2.34 e=1+11!+12!+13!+.

If za has its general value, with a0, and if w0, then

4.2.35 za=wz=exp(1aLnw).

This result is also valid when za has its principal value, provided that the branch of Lnw satisfies

4.2.36 -π(1aLnw)π.

Another example of a principal value is provided by

4.2.37 z2={z,z0,-z,z0.

Again, without the closed definition the and signs would have to be replaced by > and <, respectively.