\(\newcommand{\Neg}{\phantom{-}}\)Nowadays, mathematicians usually define a complex number \(z\) to be an ordered pair of real numbers \((x, y)\). More or less equivalently, we write \(z = x + iy\); or \(x = \operatorname{re}{z}\) and \(y = \operatorname{im}{z}\). Complex addition, multiplication, and conjugation are defined in terms of the real and imaginary parts.
At the same time, mathematicians are aware that nothing about the algebraic structure of the complex numbers tells us which square root of \(-1\) is \(i\) and which is \(-i\). Precisely, complex conjugation is an isomorphism of the complex field. This has a peculiar, and possibly under-appreciated consequence: A complex number has a real part, but does not have a well-defined imaginary part. Instead, the imaginary part is defined only up to sign. In symbols, for all real \(x\) and \(y\) we have \[ z = x + iy = x + (-y)(-i). \] We may speak of the imaginary part of \(z\) if and only if we have “picked \(i\).”
Complex Analysis
Complex analysis, at least in places, refers prominently to “the” imaginary part of a complex number. If two mathematicians develop complex analysis using different square roots of \(-1\), to what extent do their theorem statements differ? There is, after all, a sharp distinction between holomorphic and anti-holomorphic (conjugate-holomorphic) functions. Further, there are identities such as the unit circle integral \[ \oint \frac{dz}{z} = 2\pi i \] that “prefer” one square root over the other. So, what vexatious differences of signs arise between our mathematicians’ two developments? You may wish to ponder before reading further.
In An Invitation to Real Analysis, with considerations such as the preceding in mind, I wrote in a remark that all of complex analysis depends on the choice of \(i\). On the other hand, Sir Roger Penrose in The Road to Reality advocates the view that complex numbers are more physical than real numbers. That gives one pause about a passing belief that “the choice of \(i\)” matters to complex analysis.
During review of the final page proofs, having thought a bit but not enough about the issue, I hedged “all” to “much.” In fact, the sentence ought to have been struck entirely. If this is already apparent, it is safe to stop reading.
Power Series
The identity function on the complex numbers does not depend on the identification of complex numbers and ordered pairs of real numbers. The field operations are similarly independent, as is magnitude, and therefore convergence of power series. Our two mathematicians agree about which functions are analytic.
Holomorphic Functions
The usual treatment of complex differentiability starts by resolving a complex-valued function into real and imaginary parts. To check what effect the choice of imaginary unit has, let \(x\), \(y\), \(X\), and \(Y\) denote real variables, and let \(u\), \(v\), \(U\), and \(V\) denote real-valued functions on the real plane. Assume \(X = x\) and \(Y = -y\), so that \begin{align*} z &= x + iy \\ &= X – iY. \end{align*} Assume further that \(u(x, y) = U(x, -y) = U(X, Y)\) and \(v(x, y) = -V(x, -y) = -V(X, Y)\), so \begin{align*} u(x, y) + iv(x, y) &= U(x, -y) – iV(x, -y) \\ &= U(X, Y) – iV(X, Y). \end{align*} If we let \(D_{1}\) denote partial differentiation with respect to the first variable and \(D_{2}\) denote partial differentiation with respect to the second, the chain rule gives \begin{alignat*}{2} D_{1}u(x, y) &= \Neg D_{1}U(x, -y) &&= \Neg D_{1}U(X, Y), \\ D_{2}u(x, y) &= -D_{2}U(x, -y) &&= -D_{2}U(X, Y), \\ D_{1}v(x, y) &= -D_{1}V(x, -y) &&= -D_{1}V(X, Y), \\ D_{2}v(x, y) &= \Neg D_{2}V(x, -y) &&= \Neg D_{2}V(X, Y). \end{alignat*} A short calculation shows the complex-linearity equations for the function \(f(z) = u(x, y) + iv(x, y)\) are equivalent to the complex-linearity equations for \(g(z) = U(X, Y) – iV(X, Y)\); briefly, \begin{align*} D_{1}U(X, Y) &= D_{1}u(x, y) \\ &= D_{2}v(x, y) \\ &= D_{2}V(X, Y), \end{align*} and \begin{align*} D_{2}U(X, Y) &= -D_{2}u(x, y) \\ &= \Neg D_{1}v(x, Y) \\ &= -D_{1}V(X, Y). \end{align*} In words, changing the imaginary unit introduces no vexatious sign differences in the complex-linearity equations.
Students in a first course may encounter the fact that if \(f\) and \(g\) are complex-valued functions whose domains are mutually conjugate, and if \(g(z) = \overline{f(\overline{z})}\) for all \(z\) in the domain of \(g\), then \(g\) is holomorphic if and only if \(f\) is holomorphic. That is a “complex” articulation of the calculations in the preceding paragraph.
Residues
How do our two mathematicians reconcile the identity \[ \oint \frac{dz}{z} = 2\pi i, \] and more generally the calculus of residues? The right-hand side changes sign under exchange of imaginary units; either the left-hand side does as well, or else there is a vexatious sign discrepancy. In fact, the identity is the same for both mathematicians: It comes down to the meaning of counterclockwise!
A choice of imaginary unit \(i\) orients the unit circle so that the four points \(1\), \(i\), \(-1\), \(-i\) are traced in order. Alternatively, the identity \[ \exp(i\theta) = \cos\theta + i\sin\theta, \] formally the same for both choices of imaginary unit, determines opposite orientations of the unit circle. If we exchange imaginary units, this change of orientation changes the sign of the contour integral on the left. In words, integrating \(dz/z\) over the unit circle gives \(2\pi\) times the first imaginary unit along the contour.
Conclusions
Although “real” treatments of complex analysis assume a choice of imaginary unit, that choice is effectively immaterial. Our two mathematicians are looking at the same set of complex numbers, “one from each side of the plane.” While our two mathematicians orient closed paths oppositely, the respective orientations lead to identical algebraic and analytic identities. There are no vexatious differences of sign, only one complex analysis.