Chapter 3

Elementary Transcendental Functions

3.1 Exponential

Define, for all complex z, the by

∞ 1 exp(z)= ez = zn. (3.1) n! n=0 X By the ratio test, n! z z = | | 0 z, (3.2) (n + 1)!| | n +1 → ∀ the series converges everywhere. By the theorem of the Sec. 2.7, that means that the series converges uniformily in any finite closed region. Note that the following property holds:

∞ 1 exp(z + z ) = (z + z )n 1 2 n! 1 2 n=0 X∞ n 1 n! − = zmzn m n! m! (n m)! 1 2 n=0 m=0 − X∞X ∞ 1 k 1 l = z1 z2 k! ! l! ! kX=0 Xl=0 = exp(z1) exp(z2). (3.3)

Then, by induction (ez)n = enz, (3.4) where n is any positive .

23 Version of September 7, 2011 24 Version of September 7, 2011CHAPTER 3. ELEMENTARY TRANSCENDENTAL FUNCTIONS

Hyperbolic and are defined in terms of the exponen- tial function:

ez e−z ez + e−z sinh z = − cosh z = , (3.5a) 2 2 eiz e−iz eiz + e−iz sin z = − cos z = , (3.5b) 2i 2 so that

i sin z = sinh iz, (3.6a) cos z = cosh iz, (3.6b) for all complex z. Note that eiz = cos z + i sin z. (3.7)

Therefore, the polar representation of a ,

z = r(cos θ + i sin θ) = reiθ, (3.8) becomes a most useful and compact representation. In particular,

zn = rneinθ (3.9) implies De Moivre’s formula,

cos nθ + i sin nθ = (cos θ + i sin θ)n. (3.10)

3.1.1 Definition of π

There exists a positive number π such that

1. eπi/2 = i, and (3.11a)

2. ez = 1 if and only if z =2πin, (3.11b)

where n is an integer.

Hence exp(z) is periodic with period 2πi,

exp(z +2πi) = exp(z) exp(2πi) = exp(z). (3.12) 3.2. THE NATURAL 25 Version of September 7, 2011 iy z 

θ

@ @ @ @ @ x “cut” or “branch line”

Figure 3.1: Cut plane for defining the logarithm.

3.2 The

If z = reiθ, we define ln z log z ln r + iθ, (3.13) ≡ ≡ where ln r is defined as the inverse of the exponential function for real positive r, r = eln r. (3.14) Thus we have z = eζ where ζ = ln r + iθ = log z. (3.15) Recall that θ = arg z is a multivalued function, because θ is only defined up to an arbitrary multiple of 2π. [This is just the periodic property (3.12).] Re- call further that we defined the of the argument as that which satisfied π< arg z π. (3.16) − ≤ Correspondingly, we say that the single-valued logarithm function (also denoted log z) is defined in the cut plane shown in Fig. 3.1. In measuring θ from the +x axis, one is not allowed to cross the cut along the x axis. (Where the cut is placed is an arbitrary convention.) The correspondingly− defined single-valued functions arg z and log z = log z + i arg z, (3.17) | | or π< Im log z π, (3.18) − ≤ are also referred to as the principal values of the argument and logarithm, re- spectively. Now we define complex powers of complex numbers as follows:

ζz ez log ζ , (3.19) ≡ where log ζ is defined in the cut plane. Then

z ξ eξ = ez log e = ez(Re ξ+iIm ξ) = eξz (3.20)  26 Version of September 7, 2011CHAPTER 3. ELEMENTARY TRANSCENDENTAL FUNCTIONS when arg eξ = Im ξ (3.21) lies between π< Im ξ π. (3.22) − ≤ If this is not so, log eξ = ξ +2πin (3.23) where n is so chosen that

π< Im (ξ +2πin) π, (3.24) − ≤ and z eξ = ez(ξ+2πin). (3.25) For example,  1 √z = z1/2 = e 2 log z (3.26) is defined as a single-valued function only in the cut plane

π< arg z π. (3.27) − ≤ 3.3 Inverse Hyperbolic and Trigonometric Func- tions

The inverse hyperbolic and trigonometric functions are defined in terms of the logarithm:

arcsinh z = log z + (z2 + 1)1/2 , (3.28a) h i arccosh z = log z + (z2 1)1/2 , (3.28b) − 1 h 1+ z i arctanh z = log , (3.28c) 2 1 z − which are defined in the cut planes shown in Fig. 3.2.

arcsin z = i arcsinh iz − = i log iz + (1 z2)1/2 , (3.29a) − − arccos z = i arccoshh z i − = i log z + (z2 1)1/2 , (3.29b) − − arctan z = i arctanhh iz i −i 1 iz i i + z = log − = log , (3.29c) 2 1+ iz 2 i z − which are defined in the cut planes shown in Fig. 3.3. Note that the branch 3.3. INVERSE HYPERBOLIC AND TRIGONOMETRIC FUNCTIONS27 Version of September 7, 2011

arcsinh z: iy arccosh z: iy i • x iy +1 x i • •− arctanh z:

1 +1 x −• •

Figure 3.2: Cut planes for defining the inverse hyperbolic functions. The thick lines represent the cuts. arcsin z: iy arctan z: iy arccos z: +i • 1 +1 x x −• • i •−

Figure 3.3: Cut plane for defining the inverse trigonometric functions. lines (cuts) are chosen so as not to cross the region where both the range and the domain of the functions are real, because for real x,

sin x, cos x [ 1, 1], (3.30a) ∈ − tan x ( , ), (3.30b) ∈ −∞ ∞ sinh x ( , ), (3.30c) ∈ −∞ ∞ cosh x [1, ), (3.30d) ∈ ∞ tanh x [ 1, 1]. (3.30e) ∈ − An alternative notation for the inverse functions is provided by the superscript 1, as for example, − − arcsinh z = sinh 1 z, (3.31) which does not mean 1/ sinh z.