Complex Logarithms and Principal Values

In mathematics, the logarithm is the inverse operation to exponentiation. That is, the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. For example, the logarithm of 100 to base 10 is 2, because 10² = 100.

When we extend the concept of logarithms to complex numbers, things become more interesting and require careful consideration of multi-valued functions. A complex number can be represented in polar form as z = r(cos θ + i sin θ), which can also be written using Euler's formula as z = re^iθ, where r is the magnitude (or modulus) of z and θ is the argument (or phase) of z.

Definition of Complex Logarithm

Let z be a non-zero complex number. We define the complex logarithm of z, denoted as log z or ln z, as the complex number w such that e^w = z.

If we write w = u + iv, where u and v are real numbers, then e^w = e^u+iv = e^ue^iv. We also know that z = r(cos θ + i sin θ) = re^iθ. Equating these two expressions for z, we get: e^ue^iv = re^iθ

For these two complex numbers to be equal, their magnitudes must be equal and their arguments must be equal (or differ by an integer multiple of 2π). So, e^u = r and v = θ + 2kπ for some integer k.

From e^u = r, we get u = ln r, where ln r is the natural logarithm of the positive real number r. Therefore, w = u + iv = ln r + i(θ + 2kπ).

This means that for any non-zero complex number z, the complex logarithm log z is a multi-valued function, as there are infinitely many possible values for the argument θ (due to the addition of 2kπ).

The Argument of a Complex Number

The argument of a complex number z = x + iy is the angle θ it makes with the positive real axis in the complex plane. It is typically denoted as arg z. If z = r(cos θ + i sin θ), then θ = arg z. The argument is not unique; if θ is an argument, then θ + 2kπ for any integer k is also an argument. For example, if z = 1 + i, then r = |z| = sqrt(1² + 1²) = sqrt(2). The angle θ satisfies cos θ = 1/sqrt(2) and sin θ = 1/sqrt(2). So, θ = π/4 is one possible argument. Other arguments are π/4 + 2π, π/4 - 2π, etc.

Principal Value of the Complex Logarithm

To make the complex logarithm a single-valued function, we define the principal value of the logarithm. This is done by restricting the range of the argument. The principal value of the argument, denoted by Arg z, is usually chosen to be in the interval (-π, π].

If z = re^iθ, where r = |z| and θ = Arg z (the principal argument), then the principal value of the complex logarithm, denoted by Log z (with a capital 'L'), is defined as: Log z = ln r + i Arg z

Here, ln r is the standard natural logarithm of the positive real number r, and Arg z is the principal value of the argument of z, such that -π < Arg z ≤ π.

Example: Finding the Principal Value of Log(1 + i)

Let z = 1 + i. First, find the modulus r: r = |z| = sqrt(1² + 1²) = sqrt(2). Next, find the principal argument Arg z. The complex number 1 + i is in the first quadrant. The angle θ such that tan θ = 1/1 = 1 and θ is in the first quadrant is π/4. Since -π < π/4 ≤ π, this is the principal argument. Arg(1 + i) = π/4.

Now, apply the formula for the principal value of the logarithm: Log(1 + i) = ln r + i Arg z Log(1 + i) = ln(sqrt(2)) + i(π/4) Since sqrt(2) = 2^1/2, we have ln(sqrt(2)) = ln(2^1/2) = (1/2)ln(2). So, Log(1 + i) = (1/2)ln(2) + i(π/4).

Example: Finding the Principal Value of Log(-1)

Let z = -1. The modulus is r = |-1| = 1. The principal argument Arg(-1). The complex number -1 lies on the negative real axis. The angle is π. Since -π < π ≤ π, this is the principal argument. Arg(-1) = π.

Using the formula: Log(-1) = ln r + i Arg z Log(-1) = ln(1) + i(π) Since ln(1) = 0, we get: Log(-1) = 0 + iπ = iπ.

Note that the general logarithm log(-1) would be ln(1) + i(π + 2kπ) = i(π + 2kπ) for any integer k.

Example: Finding the Principal Value of Log(i)

Let z = i. The modulus is r = |i| = 1. The principal argument Arg(i). The complex number i lies on the positive imaginary axis. The angle is π/2. Since -π < π/2 ≤ π, this is the principal argument. Arg(i) = π/2.

Using the formula: Log(i) = ln r + i Arg z Log(i) = ln(1) + i(π/2) Log(i) = 0 + i(π/2) = i(π/2).

Properties of Complex Logarithms

The complex logarithm shares some properties with the real logarithm, but with important caveats due to its multi-valued nature and the principal value convention.

Logarithm of a Product

For any two non-zero complex numbers z₁ and z₂, the following relationship holds for the multi-valued logarithm: log(z₁z₂) = log z₁ + log z₂

This is because if z₁ = r₁e^iθ₁ and z₂ = r₂e^iθ₂, then z₁z₂ = r₁r₂e^{i(θ₁+θ₂)}. So, log(z₁z₂) = ln(r₁r₂) + i(θ₁ + θ₂ + 2kπ) log(z₁z₂) = ln r₁ + ln r₂ + i(θ₁ + 2k₁π) + i(θ₂ + 2k₂π) log(z₁z₂) = log z₁ + log z₂, where the total 2kπ comes from the sum of 2k₁π and 2k₂π.

However, this property does not always hold for the principal value of the logarithm: Log(z₁z₂) = Log z₁ + Log z₂ is not always true.

Let z₁ = -1 and z₂ = -1. Log z₁ = Log(-1) = iπ. Log z₂ = Log(-1) = iπ. Log z₁ + Log z₂ = iπ + iπ = 2iπ. However, z₁z₂ = (-1)(-1) = 1. Log(z₁z₂) = Log(1) = ln(1) + i Arg(1) = 0 + i(0) = 0. Clearly, 2iπ ≠ 0. The discrepancy arises because Arg(z₁) + Arg(z₂) = π + π = 2π, which is not in the range (-π, π]. The correct principal argument for z₁z₂ = 1 is 0. The relation Arg(z₁z₂) = Arg z₁ + Arg z₂ holds only up to an addition of 2π.

The correct relationship involving principal values is: Log(z₁z₂) = Log z₁ + Log z₂ + 2ikπ for some integer k. Specifically, k = 0 if -π < Arg z₁ + Arg z₂ ≤ π. k = 1 if Arg z₁ + Arg z₂ > π. k = -1 if Arg z₁ + Arg z₂ ≤ -π.

Logarithm of a Quotient

Similarly, for the multi-valued logarithm: log(z₁/z₂) = log z₁ - log z₂

For the principal value, the relationship is: Log(z₁/z₂) = Log z₁ - Log z₂ + 2ikπ for some integer k. The value of k depends on the value of Arg z₁ - Arg z₂ relative to the interval (-π, π].

Logarithm of a Power

For a complex number z and an integer n: log(zⁿ) = n log z

This property holds for the multi-valued logarithm. For the principal value, it becomes: Log(zⁿ) = n Log z + 2ikπ for some integer k. For example, Log(z²) = 2 Log z + 2ikπ. The value of k depends on 2 Arg z. If Arg z = 3π/4, then 2 Arg z = 3π/2. Arg(z²) would be 3π/2 - 2π = -π/2. So, Log(z²) would be ln(|z|²) + i(-π/2), while 2 Log z would be 2(ln|z| + i 3π/4) = 2ln|z| + i 3π/2. The difference is i(-π/2) - i(3π/2) = -2iπ, meaning k = -1.

Complex Exponentiation

Complex exponentiation is defined using the complex logarithm. For a complex base a and a complex exponent b, where a ≠ 0, we define: a^b = e^{b log a}

Since log a is multi-valued, a^b is generally a multi-valued function. If log a = ln |a| + i (Arg a + 2kπ), then a^b = e^{b (ln |a| + i (Arg a + 2kπ))} a^b = e^{b ln |a|} e^{ib (Arg a + 2kπ)} a^b = |a|^b e^{ib Arg a} e^{i 2kbπ} (assuming b is real for simplicity in notation here, but the principle applies for complex b too).

To define a unique value for a^b, we use the principal value of the logarithm, Log a: a^b = e^{b Log a} This gives the principal value of the complex exponentiation.

Example: Calculating iⁱ

We want to calculate iⁱ. Here, the base is a = i and the exponent is b = i. First, find the principal value of Log i. |i| = 1. Arg i = π/2. So, Log i = ln(1) + i(π/2) = iπ/2.

Now, use the definition a^b = e^{b Log a}: iⁱ = e^{i * Log i} iⁱ = e^{i * (iπ/2)} iⁱ = e^{i2 π/2} Since i² = -1: iⁱ = e^-π/2.

This result, e^-π/2, is a real number. It is the principal value of iⁱ.

The general value would be: iⁱ = e^{i (Log i + 2kπi)} iⁱ = e^{i (iπ/2 + 2kπi)} iⁱ = e^{i2 π/2 + 2kπi2} iⁱ = e^{-π/2 - 2kπ} = e^{-(π/2 + 2kπ)} This shows that iⁱ is a multi-valued function, with all its values being real.

Example: Calculating (-1)ⁱ

Here, a = -1 and b = i. First, find the principal value of Log(-1). |-1| = 1. Arg(-1) = π. So, Log(-1) = ln(1) + iπ = iπ.

Now, calculate the principal value of (-1)ⁱ: (-1)ⁱ = e^{i * Log(-1)} (-1)ⁱ = e^{i * (iπ)} (-1)ⁱ = e^i2π (-1)ⁱ = e^-π.

The general values are: (-1)ⁱ = e^{i (Log(-1) + 2kπi)} (-1)ⁱ = e^{i (iπ + 2kπi)} (-1)ⁱ = e^{i2π + 2kπi2} (-1)ⁱ = e^{-π - 2kπ} = e^-(1+2k)π. Again, all values are real.

Branch Cuts and Analytic Functions

For complex functions like the logarithm and powers, which are multi-valued, we often restrict the domain to make them single-valued and analytic (differentiable in the complex sense). This is achieved by introducing a branch cut.

A branch cut is a curve in the complex plane along which a given multi-valued function is discontinuous. For the complex logarithm, the standard branch cut is along the negative real axis (including the origin). This ensures that the argument Arg z is uniquely defined in the interval (-π, π].

The function Log z = ln |z| + i Arg z, with the branch cut along the negative real axis, is analytic everywhere except for z = 0 and the negative real axis itself.

If we choose a different range for the principal argument, say [0, 2π), we would place the branch cut along the positive real axis. The choice of branch cut and the corresponding principal value convention defines a specific branch of the multi-valued function. The branch defined using Arg z ∈ (-π, π] is called the principal branch.

A function is analytic if it is differentiable at every point in its domain. The complex logarithm log z is not analytic because it is multi-valued. However, by selecting a specific branch (like the principal branch using the principal value of the argument), we obtain an analytic function on a suitable domain.

Analyticity of Log z

Let f(z) = Log z = ln |z| + i Arg z. In Cartesian coordinates, if z = x + iy, then |z| = sqrt(x² + y²) and Arg z = atan2(y, x). Log z = (1/2)ln(x² + y²) + i Arg z. Let u(x, y) = (1/2)ln(x² + y²) and v(x, y) = Arg z.

For Log z to be analytic, it must satisfy the Cauchy-Riemann equations: ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x.

Let's compute the partial derivatives: ∂u/∂x = (1/2) * (1/(x²+y²)) * (2x) = x / (x² + y²) ∂u/∂y = (1/2) * (1/(x²+y²)) * (2y) = y / (x² + y²)

Now, let's consider v(x, y) = Arg z. In polar coordinates (r, θ), v = θ. The partial derivatives of θ with respect to x and y are: ∂θ/∂x = -y / (x² + y²) ∂θ/∂y = x / (x² + y²)

Comparing these: ∂u/∂x = x / (x² + y²) and ∂v/∂y = x / (x² + y²). So, ∂u/∂x = ∂v/∂y. ∂u/∂y = y / (x² + y²) and -∂v/∂x = -(-y / (x² + y²)) = y / (x² + y²). So, ∂u/∂y = -∂v/∂x.

The Cauchy-Riemann equations are satisfied. Therefore, Log z is analytic wherever Arg z is defined and differentiable, which is everywhere except for the origin and the negative real axis (where the branch cut lies).

Applications of Complex Logarithms

Complex logarithms are fundamental in various areas of mathematics, physics, and engineering.

• Solving Complex Equations: They are used to solve equations involving complex exponentials, like e^z = c or zⁿ = c. For example, solving e^z = 1 + i requires finding z = log(1 + i).
• Complex Integration: The integral of 1/z around a closed curve enclosing the origin is a classic example involving logarithms and the residue theorem.
• Conformal Mapping: Functions involving logarithms are used in conformal mapping to transform shapes and solve problems in fluid dynamics and electrostatics.
• Analysis of Singularities: Logarithmic singularities appear in solutions to differential equations and in the study of complex functions near poles or branch points.
• Control Theory: The stability of systems can be analyzed using the location of poles and zeros in the complex plane, which are related to logarithmic functions.