The exponential function The principle log Other branches

§4.1–Exponential and Logarithmic Functions

Tom Lewis

Spring Semester 2016

The exponential function The principle log Other branches Outline

The exponential function

The principle log

Other branches The exponential function The principle log Other branches Recall from §2.1 ...

Theorem Let z = x + iy and w be complex numbers. The exponential has the following properties: 1. e0 = 1. 2. ez ew = ez+w . 3. ez = ez . 4. |e(x+iy)| = ex . 1 5. = e−z . ez ez 6. = ez−w . ew 7. (ez )n = enz for each n = 0, ±1, ±2,....

The exponential function The principle log Other branches

Theorem The exponential function, ez , is entire and d ez = ez . dz The exponential function The principle log Other branches

Theorem The exponential function is periodic with period 2πi, that is,

ez+2πi = ez

for all z ∈ C.

The exponential function The principle log Other branches

Definition The strip set

S = {z = (x, y): y ∈ (−π, π], x ∈ R}

is called the fundamental region of the exponential function. The exponential function The principle log Other branches

Theorem (Geometry of the exponential function)

z 1. Let a ∈ R. The function w = e maps the vertical segment a Va = {(a, y): y ∈ (−π, π]} onto the circle {w : |w| = e }. 2. Let α ∈ (−π, π]. The function w = ez maps the horizontal line Hα = {(x, α): x ∈ R} onto the ray {w : Arg(w) = α}. 3. The function w = ez maps the fundamental strip S onto the set C \{0}.

The exponential function The principle log Other branches

Problem Given w 6= 0, find all solutions of the equation ez = w. The exponential function The principle log Other branches

Definition Logarithm Given z ∈ C \{0}, the function

ln(z) = loge (|z|) +i arg(z) real nat. log.

is called a complex logarithm| of{zz. Each} choice of arg(z) produces a branch of the logarithm function.

The exponential function The principle log Other branches

Problem Describe all possible values for ln(1 + i). The exponential function The principle log Other branches

Problem Find the solution set of the equation ez + 3 = 10e−z .

The exponential function The principle log Other branches

Theorem (Algebraic properties of ln)

Let z1, z2 ∈ C \{0} and let n ∈ Z.

• ln(z1z2) = ln(z1) + ln(z2);

• ln(z1/z2) = ln(z1) − ln(z2); n • ln(z1 ) = n ln(z1). The exponential function The principle log Other branches

Theorem 1. For z 6= 0, eln(z) = z. 2. For any z, ln(ez ) = z.

The exponential function The principle log Other branches

Definition (The principle logarithm) The function

Ln(z) = loge (|z|) + i Arg(z), for z 6= 0,

is called the principal logarithm. The exponential function The principle log Other branches

Problem Evaluate each in the form a + ib. 1. Ln(i) 2. Ln(−3) 3. Ln(−1 − i)

The exponential function The principle log Other branches

Theorem (Geometry of the principle logarithmic function)

1. Let a > 0. The function w = Ln(z) maps the circle {z : |z| = a} onto the vertical line segment

Va = {(loge (a), y): y ∈ (−π, π]}. 2. Let α ∈ (−π, π]. The function w = Ln(z) maps the ray {w : Arg(w) = α} onto the horizontal line Hα = {(x, α): x ∈ R}. 3. The function w = Ln(z) maps the set C \{0} onto the fundamental strip S. The exponential function The principle log Other branches

Definition (Principle branch of the logarithm) Let N = {(x, y): y 6 0}. The principle branch of the logarithm is Ln(z) restricted to the set

C \ N = {z : |z| > 0, −π < Arg(z) < π}.

The exponential function The principle log Other branches

Theorem The principle branch of the logarithm is continuous on its domain. The exponential function The principle log Other branches A brief digression

The CR equations in polar form Let z = (x, y) = reiθ and f (z) = u(z) + iv(z). The CR equations in polar form are: ∂u 1 ∂v ∂v −1 ∂u = , = ∂r r ∂θ ∂r r ∂θ The derivative of f at z = reiθ is ∂u ∂v  f 0(z) = e−iθ + i ∂r ∂r 1 ∂v ∂u  = e−iθ − i . r ∂θ ∂θ

The exponential function The principle log Other branches

Theorem The principal branch of the logarithm is analytic on its domain and d 1 Ln(z) = . dz z The exponential function The principle log Other branches

Problem

d 2 1. Find zez dz d 2. Find z Ln(z) dz 3. Describe all functions g such that g 0(z) = Ln(z) + 1 on the set C \ N.

The exponential function The principle log Other branches

Problem In each case, define a domain in which the given function is analytic and find its derivative. 1. Ln(z + i) 2. Ln(z2) The exponential function The principle log Other branches

Problem Define a logarithmic function according to a different branch cut and show that it is analytic in this domain.