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2025 Summer Omakase: Imaginary Numbers 📂JOF

2025 Summer Omakase: Imaginary Numbers

Introduction

We call the solution ii of the quadratic equation x2+1=0x^{2} + 1 = 0 an imaginary number in the sense of “a number in the imagination.” However, even before irrational numbers were accepted, numbers were essentially imagined entities. This season, we’ll lightly explore the world of complex numbers.

We’ve prepared the content to be deeper than a standard curriculum yet not as deep as undergraduate complex analysis.

Geometry

sinhz:=ezez2coshz:=ez+ez2 \sinh z := { {e^{z} - e^{-z}} \over 2 } \\ \cosh z := { {e^{z} + e^{-z}} \over 2 } Hyperbolic functions are usually encountered around the time of freshman undergraduate studies in the bizarre form shown above. Regardless of their unique definitions, it might be challenging to accept why they’re termed “hyperbolic sine” and “hyperbolic cosine.” sinz=eizeiz2icosz=eiz+eiz2 \sin z = { {e^{iz} - e^{-iz}} \over 2 i } \\ \cos z = { {e^{iz} + e^{-iz}} \over 2 } However, as one can infer from Euler’s formula eiθ=cosθ+isinθe^{i \theta} = \cos \theta + i \sin \theta, when observing the relationship between exponential and trigonometric functions, we see that sine and cosine can also be expressed as a linear combination of exponential functions. This perspective allows us to rationalize the nomenclature and notation of hyperbolic functions, as they perform a similar role as trigonometric functions on the imaginary axis.

Translation and Derivatives of Trigonometric Functions:

  • [1] Sine: sin(θ+n2π)=sin(n)θ\sin{(\theta +\frac { n }{ 2 }\pi )}={ \sin }^{ (n) }\theta
  • [2] Cosine: cos(θ+n2π)=cos(n)θ\cos{(\theta +\frac { n }{ 2 }\pi )}={ \cos }^{ (n) }\theta

Considering the differentiation of trigonometric functions is intriguing. By differentiating both sides of Euler’s formula with respect to θ\theta, we obtain the following:

ieiθ=(cosθ)+i(sinθ)=cos(θ+π2)+isin(θ+π2)=sinθ+icosθ \begin{align*} i e^{i \theta} =& \left( \cos \theta \right)^{\prime} + i \left( \sin \theta \right)^{\prime} \\ =& \cos \left( \theta + {\frac{ \pi }{ 2 }} \right) + i \sin \left( \theta + {\frac{ \pi }{ 2 }} \right) \\ =& -\sin \theta + i \cos \theta \end{align*}

From this result, we observe two interesting points:

  • Without differentiation, multiplying both sides by ii gives the same result as differentiation.
  • Multiplying by ii equates to a rotation of π2\frac{\pi}{2}.

This effectively reduces the relatively complex operation of differentiation to the simple operation of multiplication, demonstrating that rotation on the complex plane can also be expressed by multiplying by imaginary numbers.

Calculation

Continuing with differentiation, to numerically compute the derivative of a more general function ff, we can consider a technique called complex step differentiation. f(x)Im(f(x+ih))h f ' (x) \approx \frac{\im \left( f \left( x + i h \right) \right)}{h} Unlike the conventional mindset of “the limit of differences in function values,” this method has the advantage of requiring only a single evaluation of the function. It is derived by formally expanding the function in a Taylor series and taking only the imaginary part. f(x+ih)=f(x)+ihf(x)h2f(x)2!ih3f(x)3!s+    Imf(x+ih)=hf(x)h3f(x)3!s+    1hImf(x+ih)f(x) f \left( x + i h \right) = f (x) + i h f ' (x) - h^{2} {\frac{ f '' (x) }{ 2! }} - i h^{3} {\frac{ f ''' (x) }{ 3!s }} + \cdots \\ \implies \im f \left( x + i h \right) = h f ' (x) - h^{3} {\frac{ f ''' (x) }{ 3!s }} + \cdots \\ \implies {\frac{ 1 }{ h }} \im f \left( x + i h \right) \approx f ' (x) Since we’re interested in f(x)f ' (x), by retaining only the imaginary part and ignoring terms that can be assumed sufficiently small, we derive the desired formula. The resulting calculation yields the real-valued derivative, with the complex numbers introduced in the process vanishing.

The strategy of separating real and imaginary parts to think about problems is not difficult to find in cases seemingly unrelated to complex analysis. For instance, the following definite integral, known as a Fresnel integral, cannot be easily calculated using basic techniques. 0cosx2dx=0sinx2dx=12π2 \int_{0}^{\infty} \cos x^2 dx = \int_{0}^{\infty} \sin x^2 dx = {{1}\over{2}} \sqrt{{\pi}\over{2}} However, complex analysis allows us to convert it into a problem of contour integration, bypassing the need to directly find antiderivatives, enabling us to find the definite integral straightforwardly. To truly feel the power of complex analysis, try computing the Fresnel integral manually.

Algebra

Polynomial functions are simple and well-known, and according to Taylor’s theorem, they can approximate any continuous function, thus highlighting their importance. The most famous theorem related to polynomial functions, the Fundamental Theorem of Algebra, states that a polynomial of degree nn has exactly nn roots, counting multiplicities, a fact that is assumed as a given in almost every area of mathematics. Interestingly, most textbooks follow a complex-analytic proof rather than a purely algebraic approach to prove this theorem.

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Another instance where complex numbers appear in unrelated fields is the Gaussian integers Z[i]\mathbb{Z}[i]. These are lattice points in the complex plane, and further, the Eisenstein integers Z[ω]\mathbb{Z}[\omega] are an algebraic extension of Z\mathbb{Z} obtained from the complex root ω=e2πi/3\omega = e^{2 \pi i / 3} of x3+1x^{3} + 1.

Complex numbers not only find practical applications but also enrich the theory of mathematics itself. If you’ve only encountered practical uses of complex numbers in prerequisite courses like electronics or signal analysis, why not learn complex numbers purely for the fun of it this summer vacation?