Complex Analysis 14 | Powers [dark version]

TL;DR

For a>0, complex powers are defined by a^Z = exp(Z·ln a) using the real logarithm, so no branch ambiguity appears.

Briefing Cornell Notes

Briefing

Complex powers in the complex plane become well-defined only after choosing a consistent definition of the logarithm—and that choice determines which algebraic “power rules” remain valid. For positive real bases a, the construction starts cleanly: rational exponents are defined using roots and repeated multiplication (a^(m/n) = (n-th root of a)^m). That definition can be rewritten using real analysis via the exponential and logarithm: a^x is set to exp(x·log a), leveraging the identity log(xy)=log x + log y to justify how exponents factor out.

This approach extends from real exponents to complex exponents without changing the base: if a is a positive real number, then for any complex exponent Z, a^Z is defined as exp(Z·ln a), where ln is the real logarithm. The key point is that no ambiguity arises in the base because the logarithm of a positive real number is single-valued. As a result, expressions like e^Z are naturally interpreted as exp(Z), matching the familiar exponential function notation.

Trouble starts when the base itself is complex. Defining a^Z then requires the complex logarithm, which is inherently multi-valued because log has branch ambiguities tied to the argument (angle) of a complex number. To make a^Z single-valued, the standard move is to adopt the principal value of the complex logarithm. That restriction forces the domain to the complex plane with a slit: the negative real axis (and 0) are excluded, often described as C \ (−∞,0]. On this domain, the principal logarithm is fixed, so the principal value of the power is well-defined as exp(Z·Log a), where Log denotes the principal complex logarithm.

With the principal-value convention in place, some familiar exponent rules survive and others fail. The rule a^(Z1)·a^(Z2)=a^(Z1+Z2) still works under the principal-value definition because it aligns with the exponential law exp(u)exp(v)=exp(u+v). But the tempting rule (a^(Z1))^(Z2)=a^(Z1·Z2) generally breaks for complex exponents, since iterating complex powers interacts with branch choices of the logarithm. Integers as exponents remain safe in special cases, which is why standard real-number power laws feel reliable there.

Overall, the central takeaway is practical: complex powers are defined through exp(Z·Log a), but the logarithm’s branch choice controls the meaning of “the” power. That’s why calculations must track whether they are using principal values—some algebraic shortcuts remain valid, while others can produce incorrect results when complex arguments wrap around the branch cut.

Cornell Notes

For positive real bases a, complex powers are defined unambiguously by a^Z := exp(Z·ln a), using the real logarithm ln a. The construction begins with rational exponents via roots and repeated multiplication, then rewrites it using exp and log identities, extending to all real exponents and then to complex exponents. When the base is complex, the multi-valued complex logarithm forces a branch choice; using the principal value restricts the domain to the complex plane with a slit (excluding the negative real axis and 0). Under this principal-value convention, the exponent-addition rule a^(Z1)·a^(Z2)=a^(Z1+Z2) remains valid, but the multiplication rule (a^(Z1))^(Z2)=a^(Z1·Z2) fails in general.

How are rational exponents defined for a positive real base a, and why is that definition natural?

For a>0 and a rational exponent m/n (with n≠0), a^(m/n) is defined using roots and multiplication: a^(1/n) is the n-th root of a, and then a^(m/n) = (a^(1/n))^m. This matches the real-number intuition that exponentiation corresponds to repeated multiplication and consistent root-taking for positive bases.

What identity links exponentiation to logarithms, and how does it move the exponent into a factor?

Using the exponential/logarithm relationship, a^x is expressed as exp(x·log a). The key logarithm law is log(xy)=log x + log y, which follows from the multiplicative identity of the exponential function. With that law, repeated multiplication inside the logarithm turns into addition, so an exponent like M can be pulled out as a factor M·log(a) (and similarly for rational exponents involving roots).

Why does defining a^Z become ambiguous when the base a is complex?

The complex logarithm is multi-valued because log depends on the argument (angle) of a complex number, and angles differ by multiples of 2π. Without fixing a branch, Log a can take multiple values, so exp(Z·Log a) can produce multiple possible values for a^Z.

What does the principal value convention do, and what domain restriction results?

Choosing the principal value fixes one branch of the complex logarithm, making a^Z single-valued. That choice restricts the domain to the complex plane with a slit: the negative real axis and 0 are excluded (described as C minus the set where the principal argument jumps). On this domain, the principal logarithm Log a is well-defined, so the principal value of a^Z := exp(Z·Log a) is well-defined too.

Which complex power rules remain reliable under principal values, and which fail?

The rule a^(Z1)·a^(Z2)=a^(Z1+Z2) still holds because it corresponds to exp(u)exp(v)=exp(u+v). But (a^(Z1))^(Z2) = a^(Z1·Z2) does not hold in general for complex exponents, since iterated powers depend on how logarithm branches are handled. Integer exponents are special cases where familiar rules can still work.

Review Questions

If a is a positive real number, what exact formula defines a^Z for complex Z, and why does it avoid ambiguity?
What branch issue forces a domain restriction when defining a^Z for complex a, and what is the slit?
Give one power rule that remains valid under principal values and one that fails in general; explain the difference in terms of exponent/logarithm behavior.

Key Points

1
For a>0, complex powers are defined by a^Z = exp(Z·ln a) using the real logarithm, so no branch ambiguity appears.
2
Rational exponents m/n are first defined via roots: a^(m/n) = (a^(1/n))^m, matching real-number exponent rules for positive bases.
3
The logarithm identity log(xy)=log x + log y justifies rewriting powers as exp(x·log a) and pulling exponents into a multiplicative factor.
4
When the base a is complex, the complex logarithm is multi-valued, so a^Z is not single-valued until a branch is chosen.
5
Using the principal value fixes one logarithm branch and restricts the domain to the complex plane with a slit (excluding the negative real axis and 0).
6
Under principal values, a^(Z1)·a^(Z2)=a^(Z1+Z2) remains valid, but (a^(Z1))^(Z2)=a^(Z1·Z2) fails in general.
7
Integer exponents are safer special cases; complex-exponent power rules must be applied cautiously.

Highlights

For positive real bases, the definition a^Z := exp(Z·ln a) makes complex exponentiation straightforward and single-valued.

Principal-value complex powers require excluding the negative real axis (and 0), turning the domain into a slit plane.

The exponent-addition rule survives: a^(Z1)·a^(Z2)=a^(Z1+Z2), but exponent multiplication generally does not.

Complex powers are only as consistent as the chosen logarithm branch—algebraic shortcuts can break when branches change.

Topics

Complex Powers
Complex Logarithm
Principal Value
Exponential Function
Branch Cuts