# The complex exponential and the circle This teaching unit develops the exponential, trigonometric functions, their exact periods and polar coordinates from convergent series. Its treatment of the first zero of cosine and the unit circle is adapted from Jiří Lebl, *Basic Analysis*, volume II, version 6.3, “Complex exponential and trigonometric functions.” The product-of-series argument, explicit estimates, endpoint details and worked solutions below are editorial additions by GPT-6 Astra (OpenAI), in Codex, at Ultra, October 2026. That AI instance checked the mathematics; no human review of these additions is claimed. This unit is offered under [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/); it is a modified teaching edition, not the author's unmodified text. The [source notice](source-notice.html) identifies the supplied editable source and changes. ## Starting knowledge and destinations We use the complete ordered field of real numbers and finite complex arithmetic. The exact analytic starting proofs are in [Real analysis on closed intervals](real-analysis-on-closed-intervals.html): Proposition 1 for the Archimedean property, Lemma 2 and Theorem 3 for limits and completeness, Theorems 5–6 for extreme and intermediate values, Theorem 8 and Corollary 9 for derivatives, and Theorems 12–13 for integration. [Constructing the real numbers](constructing-the-real-numbers.html) supplies the ordered-field construction and completeness in Theorems 4, 5 and 7, and uniqueness in Theorem 10. Rational arithmetic, induction and elementary set constructions are the stated starting inputs of that preceding unit. The estimates below supply the series operations needed here; neither Cauchy's integral theorem nor the complex identity theorem is an input. The results supply the exponential and angle facts used in [Cauchy's theorem for cycles](../../courses/foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences.html) and in the winding argument for the Bott projection. In particular, knowing that the circle is parametrized periodically is not the same as proving that its period is exactly \(2\pi\). ## Convergent series and the exponential **Lemma 1 (Absolute convergence and multiplication).** For every real \(R\geq0\), the series \(\sum_{n\geq0}R^n/n!\) converges. For complex \(z,w\), the two series with terms \(z^n/n!\) and \(w^n/n!\) may be multiplied and grouped by total degree. **Proof.** Choose an integer \(N\geq 2R\). For \(n\geq N\), successive terms have ratio \(R/(n+1)\leq1/2\); thus every tail is bounded by a convergent geometric series. This proves convergence, uniformly for \(|z|\leq R\). More generally, if \(\sum_j |a_j|=A<\infty\) and \(\sum_k |b_k|=B<\infty\), the sum of \(|a_jb_k|\) over any finite rectangle is at most \(AB\). The sum over pairs with \(j>M\) or \(k>M\) is at most \[ B\sum_{j>M}|a_j|+A\sum_{k>M}|b_k|, \] which tends to zero. All finite sets of pairs containing a sufficiently large square therefore give the same limit. Both rectangular summation and grouping by total degree exhaust such squares, so both have that limit. Finite rectangular sums are products of finite sums; taking their limits proves the multiplication assertion. \(\square\) Define \[ E(z)=\sum_{n=0}^{\infty}\frac{z^n}{n!},\qquad z\in\mathbb C. \] **Theorem 2 (Addition, inversion and differentiation).** For all complex \(z,w\), \[ E(0)=1,\quad E(z+w)=E(z)E(w),\quad E(z)\ne0, \quad E(-z)=E(z)^{-1},\quad E'(z)=E(z), \quad \overline{E(z)}=E(\bar z). \] For real \(x\), \(E(x)>0\). Its restriction to the real line is strictly increasing and maps that line onto \((0,\infty)\). **Proof.** The binomial theorem and Lemma 1 give \[ E(z)E(w)=\sum_{n=0}^{\infty}\sum_{j=0}^{n} \frac{z^jw^{n-j}}{j!(n-j)!} =\sum_{n=0}^{\infty}\frac{(z+w)^n}{n!}=E(z+w). \] Taking \(w=-z\) proves inversion and nonvanishing. Conjugation commutes with limits of finite sums and gives the last identity. Let \(C=\sum_{n=2}^{\infty}1/n!<\infty\). If \(|h|\leq1\), \[ |E(h)-1-h|\leq C|h|^2. \] Consequently \(E(h)\to1\) and \((E(h)-1)/h\to1\). The addition law shows that \(E\) is continuous everywhere and that \[ \frac{E(z+h)-E(z)}h =E(z)\frac{E(h)-1}h\longrightarrow E(z). \] For real \(x\), the series is real and \(E(x)=E(x/2)^2>0\). Its derivative is therefore positive, so the mean value theorem gives strict increase. For \(x\geq0\), the series gives \(E(x)\geq1+x\); hence \(E(x)\to\infty\) as \(x\to\infty\). Inversion then gives \(E(x)\to0\) as \(x\to-\infty\). Continuity and the intermediate value theorem prove surjectivity onto \((0,\infty)\). \(\square\) We write \(e^z=E(z)\). This agrees with the real exponential characterized by \(f'=f\), \(f(0)=1\): if a real differentiable \(f\) has those properties, differentiation gives \((f(x)E(-x))'=0\), so the product is identically one. Define \(\log r\), for \(r>0\), as the unique real \(x\) with \(E(x)=r\). ## Sine, cosine and elementary identities For complex \(z\), define \[ \cos z=\frac{E(iz)+E(-iz)}2,\qquad \sin z=\frac{E(iz)-E(-iz)}{2i}. \] **Theorem 3 (Trigonometric identities).** For every complex \(z\), \[ E(iz)=\cos z+i\sin z,\qquad \cos^2z+\sin^2z=1, \] \[ \cos(-z)=\cos z,\quad\sin(-z)=-\sin z, \quad\cos'(z)=-\sin z,\quad\sin'(z)=\cos z. \] Their values at zero are \(1,0\), respectively, and \[ \cos z=\sum_{n=0}^{\infty}\frac{(-1)^nz^{2n}}{(2n)!},\qquad \sin z=\sum_{n=0}^{\infty}\frac{(-1)^nz^{2n+1}}{(2n+1)!}. \] For real \(x\), sine and cosine are real, \(E(ix)\) has modulus one, \(|\sin x|,|\cos x|\leq1\), and \(\sin x\leq x\) when \(x\geq0\). **Proof.** Euler's identity, the values at zero and parity follow by adding, subtracting or reversing the defining expressions. Set \(a=E(iz)\), \(b=E(-iz)\). Then \(ab=1\), and \[ \left(\frac{a+b}{2}\right)^2+ \left(\frac{a-b}{2i}\right)^2=ab=1. \] This proves the square identity for complex as well as real arguments. Theorem 2 and the chain rule give \[ \cos'(z)=\frac{iE(iz)-iE(-iz)}2=-\sin z, \quad \sin'(z)=\frac{iE(iz)+iE(-iz)}{2i}=\cos z. \] In the two defining series, even powers agree and odd powers cancel when adding; odd powers agree and even powers cancel when subtracting and dividing by \(i\). Absolute convergence justifies these operations and gives the displayed series. For real \(x\), \(E(-ix)=\overline{E(ix)}\). Thus cosine and sine are its real and imaginary parts, while \(E(ix)\overline{E(ix)}=1\). Their absolute values are at most one. Finally \((x-\sin x)'=1-\cos x\geq0\) and the difference vanishes at zero. The real mean value theorem proves the last inequality. \(\square\) Multiplication of \(E(iz)\) and \(E(iw)\) gives, by the definitions, the addition identities \[ \cos(z+w)=\cos z\cos w-\sin z\sin w, \quad \sin(z+w)=\sin z\cos w+\cos z\sin w. \] These are identities for complex arguments; they do not require a geometric definition of angle. ## The exact period and polar coordinates **Theorem 4 (The first zero and the circle).** Cosine has a smallest positive zero \(r\). Define \(\pi=2r\). Then \[ E(i\pi/2)=i,\quad E(i\pi)=-1,\quad E(2\pi i)=1. \] The map \(t\mapsto E(it)\) is a bijection from \([0,2\pi)\) onto the unit circle. Moreover \[ E(it)=1\quad\Longleftrightarrow\quad t\in2\pi\mathbb Z \qquad(t\in\mathbb R). \] Both sine and cosine have least positive real period \(2\pi\), even when regarded as functions on \(\mathbb C\). **Proof.** Cosine is positive near zero. If it had no positive zero, continuity would force it to be positive everywhere on \([0,\infty)\). Sine would then be strictly increasing there. Fix \(a>0\); then \(\sin a>0\), and for \(t>a\) the fundamental theorem gives \[ \cos t=\cos a-\int_a^t\sin s\,ds \leq\cos a-(t-a)\sin a, \] which is eventually negative. This contradiction proves existence of a positive zero. If \(T\) is such a zero, the zeros in \([0,T]\) form a nonempty closed set separated from zero. Its infimum is positive and is attained by continuity; call it \(r\). Cosine is positive on \([0,r)\). Sine is strictly increasing there, so \(\sin r>0\). The square identity and \(\cos r=0\) imply \(\sin r=1\). Hence \(E(ir)=i\), and the addition law gives \(E(2ir)=-1\) and \(E(4ir)=1\). No \(t\in(0,4r)\) satisfies \(E(it)=1\). Indeed, \(u=E(it/4)\) has strictly positive real and imaginary parts, since \(00\) is a period of cosine, then \(\cos T=1\), so the real square identity gives \(\sin T=0\), and \(E(iT)=1\). If \(T>0\) is a period of sine, then \(\sin T=0\) and \[ 1=\sin(r+T)=\sin r\cos T+\cos r\sin T=\cos T. \] Again \(E(iT)=1\). The kernel assertion proves that no smaller positive period exists. \(\square\) **Corollary 5 (The exponential fibre and polar form).** Every nonzero complex \(w\) can be written \[ w=|w|E(i\theta),\qquad 0\leq\theta<2\pi. \] This \(\theta\) is unique. All solutions of \(E(z)=w\) are \[ z=\log|w|+i(\theta+2\pi n),\qquad n\in\mathbb Z. \] In particular \(E(z)=E(w)\) if and only if \(z-w\in2\pi i\mathbb Z\). **Proof.** Apply Theorem 4 to \(w/|w|\). For real \(x,y\), the addition law and unit modulus give \[ |E(x+iy)|=E(x),\qquad E(x+iy)=E(x)E(iy). \] Therefore any solution has \(x=\log|w|\), and its imaginary part is precisely one of the angles supplied by the kernel assertion. The final equivalence follows by applying this to \(E(z-w)=1\). \(\square\) **Proposition 6 (A continuous local angle).** On the open right semicircle, there is a unique continuous angle \(\alpha(u)\in(-\pi/2,\pi/2)\) with \(E(i\alpha(u))=u\). Consequently any circle-valued continuous function whose values lie in an open semicircle has a continuous real argument there, after multiplying by a fixed unit scalar. **Proof.** Sine is strictly increasing on \([-r,r]\): its derivative is positive in the interior, and continuity gives the endpoint inequalities. It maps that interval onto \([-1,1]\). Its inverse is continuous. To check the latter directly at a value corresponding to \(t\in(-r,r)\), choose \(\varepsilon>0\) with \(t\pm\varepsilon\in(-r,r)\). The strict inequalities \[ \sin(t-\varepsilon)<\sin t<\sin(t+\varepsilon) \] force all sufficiently nearby values to have inverse in \((t-\varepsilon,t+\varepsilon)\); the endpoint proof is one-sided. Define \(\alpha(u)\) to be that inverse at \(\operatorname{Im}u\). For \(\operatorname{Re}u>0\), it lies in \((-r,r)\); its sine is the given imaginary part and its positive cosine is \(\sqrt{1-(\operatorname{Im}u)^2}=\operatorname{Re}u\). This proves both the formula and continuity. Uniqueness follows from Theorem 4. More explicitly, if the semicircle is \(c\{v:|v|=1,\operatorname{Re}v>0\}\) and \(E(i\theta_c)=c\), then \(\theta_c+\alpha(\bar c\,u)\) is its continuous argument. \(\square\) ## Worked exercises **Exercise 1.** Derive the power series for sine and cosine and their real derivatives without assuming trigonometric formulas. **Solution.** In \(E(iz)+E(-iz)\), the coefficient of \(z^{2n}\) is \(2(-1)^n/(2n)!\) and every odd coefficient vanishes. Dividing by two gives cosine. In the difference, the coefficient of \(z^{2n+1}\) is \(2i(-1)^n/(2n+1)!\); division by \(2i\) gives sine. Their derivatives follow by differentiating the defining exponential expressions using Theorem 2 and the real chain rule, exactly as in Theorem 3. This argument does not assume a rule for differentiating an arbitrary infinite series. \(\square\) **Exercise 2.** Find every solution of \(E(z)=-2\). **Solution.** The modulus equation gives \(\operatorname{Re}z=\log2\). The argument of \(-1\) in \([0,2\pi)\) is \(\pi\), so Corollary 5 gives \(z=\log2+i(2n+1)\pi\), for \(n\in\mathbb Z\), and substitution verifies every value. \(\square\) **Exercise 3.** For an integer \(m\geq1\), list all \(m\)-th roots of a nonzero complex number \(w=\rho E(i\theta)\), where \(\rho>0\). **Solution.** Set \(a=E((\log\rho)/m)>0\). Then \(a^m=\rho\). The \(m\) numbers \[ aE\left(i\frac{\theta+2\pi k}{m}\right),\qquad k=0,\ldots,m-1, \] have \(m\)-th power \(w\). They are distinct because the difference between two listed angles is a multiple of \(2\pi\) only when their indices agree. Conversely any root has modulus \(a\), by positivity and strict increase of the real exponential. Writing that root in polar form, its angle satisfies \(m\varphi-\theta\in2\pi\mathbb Z\); reduction of the integer modulo \(m\) puts it on the list. \(\square\) **Exercise 4.** Prove that every continuous \(u:[0,1]\to\mathbb C\) with \(|u(t)|=1\) has a continuous argument, and that a chosen value at \(0\) determines the argument uniquely. **Solution.** Continuity on the compact interval implies uniform continuity. Choose a finite partition \(0=t_0<\cdots