Elementary Complex Functions

Extend a handful of real functions to the complex plane and something remarkable happens: they collapse into one. On the real line the exponential, the sine, the cosine, the hyperbolic functions, the logarithm and the power all look like separate species, each with its own graph and its own identities. Allow the variable to be complex and a single entire function — the exponential $e^z$ — turns out to generate essentially all of them. Trigonometric and hyperbolic functions are just linear combinations of $e^{iz}$ and $e^{-iz}$ ; the logarithm and general powers are its inverses. Oscillation, exponential growth, and hyperbolic geometry are revealed as three cross-sections of one analytic object. The economy comes at a price, and the price is multivaluedness. Because the exponential repeats itself in the imaginary direction — $e^{z+2\pi i}=e^z$ — it cannot be undone by any single-valued function. Its inverse, the logarithm, returns an answer only up to additive multiples of $2\pi i$ , and to pin down one value you must cut the plane and forbid crossing the cut. That branch structure propagates into every power $z^a$ and into the inverse trigonometric functions, and is the physicist's first genuine encounter with the Riemann surface. These functions are the working vocabulary of everything downstream. Taylor and Laurent series expand them; the residue calculus integrates them; poles, zeros, and analytic continuation all assume fluency with $e^z$ , $\log z$ , $\sin z$ and their kin. This article defines them, derives the identities that bind the families together, and maps out the branch behaviour that makes the subject subtle.

Key takeaways

The complex exponential $e^z=e^x(\cos y+i\sin y)$ is the master function — entire, nowhere zero, and periodic with imaginary period $2\pi i$ — from which every other elementary function is built.
Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ fuses the trigonometric and exponential worlds, so $\cos$ , $\sin$ , $\cosh$ , $\sinh$ are four faces of one object with $\cos(iz)=\cosh z$ and $\sin(iz)=i\sinh z$ .
The logarithm $\log z=\ln|z|+i\arg z$ is multivalued because $\arg z$ is fixed only modulo $2\pi$ ; choosing the principal branch requires cutting the plane along the negative real axis.
General powers $z^a=e^{a\log z}$ inherit the ambiguity: single-valued for integer $a$ , finitely multivalued for rational $a$ , infinitely multivalued otherwise — so $i^i=e^{-\pi/2}\approx0.2079$ is one of infinitely many real values.
Complex sine and cosine are unbounded, $|\sin z|\sim\tfrac12 e^{|y|}$ off the real axis, and the inverse trig and inverse hyperbolic functions are logarithms in disguise.
Key idea: everything follows from one entire periodic function, $e^z=e^x(\cos y+i\sin y)$ ; the trigonometric and hyperbolic functions are its linear combinations, the logarithm and powers are its multivalued inverses, and periodicity in the imaginary direction is exactly what forces that multivaluedness.

The core insight

Everything follows from one entire periodic function, $e^z=e^x(\cos y+i\sin y)$ . Trigonometric and hyperbolic functions are linear combinations of exponentials; the logarithm and powers are its multivalued inverses. Periodicity in the imaginary direction is precisely what forces the logarithm and powers to be multivalued.

Where this lives today

The elementary complex functions are not a historical set piece; they are the daily arithmetic of physics and engineering. Every alternating-current network is solved by writing a sinusoid $V\cos(\omega t+\varphi)$ as $\operatorname{Re}(Ve^{i\varphi}e^{i\omega t})$ , so a differential equation for an $RLC$ circuit collapses to algebra in $\mathbb{C}$ — the phasor method behind every power grid and radio. In quantum mechanics the time-evolution factor $e^{-iEt/\hbar}$ and the spin rotation $e^{-i\theta\,\hat n\cdot\boldsymbol\sigma/2}=\cos\tfrac\theta2-i\,\hat n\cdot\boldsymbol\sigma\sin\tfrac\theta2$ are Euler's formula generalized to operators, and their phase produces every interference effect from the double slit to the Aharonov–Bohm shift. The residue calculus that evaluates real integrals like $\int_0^{2\pi}\!\tfrac{\mathrm{d}\theta}{a+b\cos\theta}$ via $z=e^{i\theta}$ rests on the exponential form of the trig functions. And branch cuts are physical: the cut of $\sqrt{k^2-\kappa^2}$ enforces outgoing versus decaying waves in scattering, and the analytic structure of response functions in the complex frequency plane underlies the dispersion relations and Kramers–Kronig identities of optics and particle physics.

The physical picture

Read $e^z$ as a machine with two knobs. The real part $x$ sets a scale — the output has modulus $e^x$ — and the imaginary part $y$ sets a rotation, pointing the output at angle $y$ . So a vertical line $x=\text{const}$ is wound into a circle of radius $e^x$ , and a horizontal line $y=\text{const}$ is stretched into a ray at angle $y$ . Because $y$ and $y+2\pi$ name the same direction, the exponential wraps the plane around the origin infinitely many times — and that wrapping is the single source of every multivaluedness that follows. The first figure shows the fundamental strip $-\pi<y\le\pi$ mapped onto the whole punctured plane exactly once.

Figure

The exponential map w = eᶻ. A vertical line x = const becomes a circle of radius eˣ, so x = 0 maps to the unit circle and x = 1 to a circle of radius e¹ = 2.718. The fundamental strip from −π to π covers the whole punctured plane once, and the marked point z = 1 + iπ⁄2 lands on w = e·e^(iπ⁄2) = 2.718 i

The logarithm undoes the wrapping, but only after you decide where to cut. To flatten the infinitely wound plane back to a single sheet you must slice it open along some ray, the standard choice being the negative real axis. On the cut plane the principal logarithm assigns an imaginary part in $(-\pi,\pi]$ ; step across the cut and it jumps by $2\pi$ , exactly like climbing a spiral staircase and arriving one floor above where you started. The second figure makes this branch structure concrete, with the checkpoint value $\operatorname{Log}(-1)=i\pi$ .

Figure

The principal logarithm and its branch cut. Log z = ln r + i Arg z with Arg z ranging over −π to π, cut along the negative real axis. Just above the cut Arg → +π, just below Arg → −π, a jump of 2π; at z = −1 the principal value is Log(−1) = iπ ≈ 3.1416 i, and the full multivalued log adds 2πik for every integer k

Finally, the trigonometric functions are the imaginary-axis relatives of the hyperbolic ones: $\cos(iy)=\cosh y$ and $\sin(iy)=i\sinh y$ , so oscillation along the real axis becomes exponential growth along the imaginary axis. That single fact — that $\sin z$ and $\cos z$ blow up like $e^{|y|}$ off the real line rather than staying trapped in $[-1,1]$ — is why the complex trig functions are unbounded and why estimating contour integrals (Jordan's lemma) demands care about which way one closes.

The core equations

The whole subject is anchored by one definition. Demand an entire function that agrees with $e^x$ on the real axis and obeys $f'(z)=f(z)$ ; the everywhere-convergent power series $\sum z^n/n!$ delivers it, and splitting the argument into real and imaginary parts gives the modulus–argument form.

e^{z}=e^{x}\left(\cos y+i\sin y\right),\qquad z=x+iy \qquad (1)

where: $z=x+iy$ — complex argument with $x,y\in\mathbb{R}$ ; $e^{x}$ — real exponential, fixing the modulus $|e^z|=e^x$ ; $\cos y,\ \sin y$ — real trig functions, fixing the argument $\arg e^z=y$ . Reading: the real part scales, the imaginary part rotates, and since $e^x>0$ , $e^z$ is never zero. The bridge between the exponential and trigonometric worlds is Euler's formula, obtained below by sorting the exponential series into even and odd powers. It is the identity the whole chapter leans on.

e^{i\theta}=\cos\theta+i\sin\theta \qquad (2)

where: $\theta$ — real angle in radians; $e^{i\theta}$ — unit-modulus number rotating by $\theta$ . Reading: a point on the unit circle is the same thing as an exponential of an imaginary angle, so $\theta=\pi$ gives the celebrated $e^{i\pi}=-1$ . Inverting Euler's formula solves for the trig functions in terms of exponentials. These definitions extend $\cos$ and $\sin$ to all complex arguments while reproducing the real functions on the axis.

\cos z=\frac{e^{iz}+e^{-iz}}{2},\qquad \sin z=\frac{e^{iz}-e^{-iz}}{2i} \qquad (3)

where: $z$ — complex argument; $e^{\pm iz}$ — the two exponential building blocks. Reading: every trigonometric identity is now a statement about exponentials, provable by the law $e^{a}e^{b}=e^{a+b}$ alone. The hyperbolic functions are the same construction without the factors of $i$ , which is exactly why they are the trig functions rotated onto the imaginary axis.

\cosh z=\frac{e^{z}+e^{-z}}{2},\qquad \sinh z=\frac{e^{z}-e^{-z}}{2},\qquad \cos(iz)=\cosh z,\ \ \sin(iz)=i\sinh z \qquad (4)

where: $\cosh z,\ \sinh z$ — hyperbolic cosine and sine. Reading: the bridge identities convert any hyperbolic fact into a trigonometric one and back, so the two families are literally one family seen along two axes. The inverse of the exponential is the logarithm, and because the exponential is periodic the logarithm is irreducibly multivalued — it carries a free integer $k$ labelling which sheet of the Riemann surface you are on.

\log z=\ln|z|+i\arg z=\ln|z|+i(\operatorname{Arg}z+2\pi k),\qquad k\in\mathbb{Z} \qquad (5)

where: $\ln|z|$ — real logarithm of the modulus $|z|=r>0$ ; $\operatorname{Arg}z$ — principal argument with $-\pi<\operatorname{Arg}z\le\pi$ ; $k$ — integer branch index. Reading: the modulus fixes the real part uniquely, but the argument is free modulo $2\pi$ , so $\log$ returns infinitely many values differing by $2\pi i$ . Selecting $k=0$ and cutting the negative real axis produces the principal logarithm, single-valued and analytic on the slit plane.

\operatorname{Log}z=\ln|z|+i\operatorname{Arg}z,\qquad -\pi<\operatorname{Arg}z\le\pi \qquad (6)

where: $\operatorname{Log}z$ — the principal branch, analytic on $\mathbb{C}\setminus(-\infty,0]$ with derivative $1/z$ . Reading: this is the value a calculator returns, but the cut means $\operatorname{Log}$ jumps by $2\pi i$ as you cross the negative real axis. General powers are defined through the logarithm, so they inherit its branch structure exactly. The count of distinct values is decided entirely by the arithmetic nature of the exponent.

z^{a}=e^{a\log z},\qquad a\in\mathbb{C},\ z\neq0 \qquad (7)

where: $a$ — complex exponent; $\log z$ — the multivalued logarithm. Reading: integer $a$ collapses the ambiguity to a single value, rational $a=p/q$ leaves $q$ values, and irrational or genuinely complex $a$ leaves infinitely many — which is how $i^i$ becomes an infinite ladder of real numbers. The derivatives close the toolkit and confirm that these functions are analytic where expected — entire for the exponential and trig families, and analytic on the cut plane for the logarithm.

\frac{\mathrm{d}}{\mathrm{d}z}e^{z}=e^{z},\qquad \frac{\mathrm{d}}{\mathrm{d}z}\sin z=\cos z,\qquad \frac{\mathrm{d}}{\mathrm{d}z}\operatorname{Log}z=\frac1z \qquad (8)

where: $\tfrac{\mathrm{d}}{\mathrm{d}z}$ — the complex derivative, defined for analytic functions. Reading: the rules match the real case exactly because each function is given by a power series, and $\tfrac{\mathrm{d}}{\mathrm{d}z}\operatorname{Log}z=1/z$ is the seed of the entire residue calculus.

Derivation AEuler's formula from the exponential series

The goal is to prove Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ using nothing but the power series that defines the exponential, so that the identity rests on algebra alone and assumes no geometry and no prior knowledge that $e^{i\theta}$ lives on the unit circle. We insert an imaginary argument into the everywhere-convergent series, exploit the four-fold cycle of the powers of $i$ to sort the terms by parity, and recognize the two resulting real series as the Maclaurin expansions of cosine and sine.

Step 1. Write the exponential as its power series and substitute $w=i\theta$ with $\theta$ real: $e^{i\theta}=\sum_{n=0}^{\infty}(i\theta)^n/n!$ . Why: the series is the only definition of $e^w$ that survives the passage from real to complex argument unchanged, so it is the safe bridge into $\mathbb{C}$ , and its absolute convergence permits the reordering to come.

Step 2. Use the cycle of the powers of $i$ , namely $i^0=1,\ i^1=i,\ i^2=-1,\ i^3=-i,\ i^4=1,\dots$ , which repeats with period four. Why: this four-fold pattern is precisely the mechanism that separates the sum into a purely real channel and a purely imaginary channel, sorting even powers from odd.

Step 3. Collect the even terms $n=2m$ (real, with sign $(-1)^m$ ) apart from the odd terms $n=2m+1$ (imaginary), giving $e^{i\theta}=\sum_{m=0}^{\infty}\frac{(-1)^m\theta^{2m}}{(2m)!}+i\sum_{m=0}^{\infty}\frac{(-1)^m\theta^{2m+1}}{(2m+1)!}$ . Why: the parity split is exactly the content of the identity — one channel will be cosine, the other sine, and nothing else is needed.

Step 4. Recognize the two series as the standard Maclaurin expansions $\cos\theta=\sum(-1)^m\theta^{2m}/(2m)!$ and $\sin\theta=\sum(-1)^m\theta^{2m+1}/(2m+1)!$ , so $e^{i\theta}=\cos\theta+i\sin\theta$ . Why: identifying the textbook series completes the proof with no appeal to geometry; the geometry of the unit circle is a consequence, not an assumption. *As a check: setting $\theta=\pi$ gives $e^{i\pi}=\cos\pi+i\sin\pi=-1$ , i.e. $e^{i\pi}+1=0$ .*

Go deeper · Derivation B · 1 minThe complex logarithm as the inverse exponential

Derivation BThe complex logarithm as the inverse exponential

The aim is to construct $\log z$ by solving $e^{w}=z$ directly, and in doing so to expose why the answer cannot be single-valued: the exponential's imaginary periodicity leaves the argument undetermined modulo $2\pi$ , so inverting it necessarily produces an infinite family of values differing by $2\pi i$ . We split $w$ into real and imaginary parts, match modulus and argument separately, and read off both the multivalued logarithm and the principal branch a cut recovers.

Step 1. Write the unknown as $w=u+iv$ and the target in polar form $z=re^{i\theta}$ with $r=|z|>0$ , so that $e^{w}=z$ becomes $e^{u}e^{iv}=re^{i\theta}$ . Why: polar form separates a complex equation into an independent length equation and angle equation, which decouple cleanly and can be solved one at a time.

Step 2. Match the moduli: $e^{u}=r$ forces $u=\ln r=\ln|z|$ , the ordinary real logarithm of a positive number. Why: the modulus is a positive real, where the real logarithm is unique, so no ambiguity can enter through the real part of $w$ .

Step 3. Match the arguments, but only up to whole turns: $v=\theta+2\pi k$ for every integer $k$ , because $e^{iv}$ is unchanged by adding $2\pi$ to $v$ . Why: this is the exact point where multivaluedness is born — equality of two directions holds only modulo a full revolution, and each $k$ is a distinct sheet of the Riemann surface.

Step 4. Assemble $\log z=\ln|z|+i(\operatorname{Arg}z+2\pi k)$ , and recover a single-valued function by fixing $k=0$ and demanding $-\pi<\operatorname{Arg}z\le\pi$ , which is the principal branch $\operatorname{Log}z$ analytic on the plane cut along $(-\infty,0]$ . Why: a branch cut is not optional bookkeeping but the price of turning a multivalued relation into an honest analytic function, and the cut is where $\operatorname{Log}$ is discontinuous by $2\pi i$ . *As a check: $\operatorname{Log}(-1)=\ln 1+i\pi=i\pi$ since $|-1|=1$ and $\operatorname{Arg}(-1)=\pi$ ; the full multivalued set is $i\pi+2\pi ik$ .*

Go deeper · Derivation C · 1 minThe inverse sine expressed as a logarithm

Derivation CThe inverse sine expressed as a logarithm

The goal is to show that $\arcsin$ , which on the real line looks like an ordinary bounded function, is in truth a logarithm carrying square-root branch points at $z=\pm1$ . Inverting $\sin w=z$ with the exponential definition turns the equation into a quadratic in $e^{iw}$ ; solving it introduces a two-valued square root, and undoing the exponential introduces a logarithm, so the multivaluedness of the inverse sine is assembled explicitly from these two familiar sources.

Step 1. Start from the definition $\sin w=z$ , i.e. $\dfrac{e^{iw}-e^{-iw}}{2i}=z$ , writing everything in exponentials. Why: the exponential form is what converts a transcendental equation into an algebraic one, the essential move that makes the inversion tractable.

Step 2. Set $\zeta=e^{iw}$ and clear the denominator: $\zeta-\zeta^{-1}=2iz$ becomes the quadratic $\zeta^{2}-2iz\,\zeta-1=0$ . Why: substituting $\zeta=e^{iw}$ collapses the two exponentials $e^{\pm iw}$ into one unknown and its reciprocal, and multiplying through by $\zeta$ produces a plain quadratic.

Step 3. Solve by the quadratic formula: $\zeta=iz\pm\sqrt{-z^{2}+1}=iz+\sqrt{1-z^{2}}$ , since the discriminant $\sqrt{(2iz)^2+4}=2\sqrt{1-z^2}$ and the two-valued root already absorbs the $\pm$ . Why: the square root is where the branch points at $z=\pm1$ enter — there the quantity under the root vanishes and the two roots collide.

Step 4. Undo the exponential: $iw=\log\zeta$ gives $w=\arcsin z=-i\log\!\left(iz+\sqrt{1-z^{2}}\right)$ , multivalued through both the root and the log. Why: the logarithm supplies the second, additive layer of multivaluedness on top of the sign choice in the root, which together give $\arcsin$ its full branch structure. *As a check: at $z=0$ , $\arcsin 0=-i\log(0+1)=0$ , matching $\sin 0=0$ ; and $\arcsin 2=\tfrac{\pi}{2}+i\,\ln(2+\sqrt3)=\tfrac{\pi}{2}+1.3170\,i$ on the principal branch, a genuinely complex value because $2$ lies outside $[-1,1]$ .*

Limiting cases

The complex functions must reduce to the real ones on the axis and diverge from them off it; the table records both behaviours.

Case	Result	Meaning
Real argument	$e^{x},\ \sin x,\ \ln x\ (x>0)$	Setting $y=0$ (or $z=x>0$ for the log) recovers exactly the real elementary functions; the principal branch agrees with the real value.
Purely imaginary argument	$e^{iy}=\cos y+i\sin y,\ \ \cos(iy)=\cosh y$	Euler's formula and the trig–hyperbolic bridge: oscillation and exponential growth swap roles under $x\leftrightarrow iy$ .
Integer power	$z^{n}=e^{n\log z}$ single-valued	The $2\pi i n$ ambiguity in $\log z$ contributes $e^{2\pi i n k}=1$ , collapsing the values to one; non-integer exponents stay multivalued.
Large imaginary part	$\lvert\sin z\rvert\sim\tfrac12 e^{\lvert y\rvert}$ as $\lvert y\rvert\to\infty$	Unlike on the real line, complex sine and cosine are unbounded — the key fact behind Jordan's lemma and contour estimates.

Common trapReal-variable identities for log and powers carry over unchanged

They do not, and assuming so is the classic complex-analysis blunder. The identities $\log(z_1z_2)=\log z_1+\log z_2$ and $(z_1z_2)^a=z_1^a z_2^a$ FAIL for the principal branch — they hold only up to an added multiple of $2\pi i$ or a branch-dependent factor, i.e. only as equalities of full multivalued sets. Likewise $\operatorname{Log}(e^z)=z$ only when $-\pi<\operatorname{Im}z\le\pi$ , and $\sqrt{z_1}\sqrt{z_2}=\sqrt{z_1z_2}$ can fail by a sign. The notorious consequence of ignoring this is the false chain $1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}=i\cdot i=-1$ . Always track which branch you are on, and never assume a real identity survives the crossing of a cut.

Worked examples

1 · The infinitely many values of $i^i$ (confidence builder)

Evaluate $i^i$ . Show it is real, find its principal value, and list all of its values.

Step 1. Use the definition of a general power, $i^i=e^{i\log i}$ . The logarithm of $i$ is needed, and it is multivalued: $|i|=1$ so $\ln|i|=0$ , and $\arg i=\tfrac{\pi}{2}+2\pi k$ , giving $\log i=i\left(\tfrac{\pi}{2}+2\pi k\right)$ for every integer $k$ .

Step 2. Substitute: $i^i=e^{\,i\cdot i(\pi/2+2\pi k)}=e^{-(\pi/2+2\pi k)}$ . The two factors of $i$ multiply to $-1$ , so the exponent is real and negative — every value of $i^i$ is a positive real number, a small shock the first time one sees it.

Step 3. The principal value takes $k=0$ : $i^i=e^{-\pi/2}=\mathbf{0.20788}$ (to five figures). Other branches give $e^{-\pi/2-2\pi k}$ , so $k=-1$ yields $e^{-\pi/2+2\pi}\approx 111.32$ and $k=1$ yields $e^{-\pi/2-2\pi}\approx 3.882\times10^{-4}$ .

Step 4. Because the exponent $i$ is not rational, no two branches coincide: there are infinitely many distinct real values, a geometric ladder with ratio $e^{-2\pi}\approx1.87\times10^{-3}$ . $i^i$ is real, its principal value is $e^{-\pi/2}\approx0.2079$ , and it has infinitely many values $e^{-\pi/2-2\pi k}$ .

2 · Complex sine off the real axis (fully worked)

Evaluate $\sin(1+i)$ numerically, and confirm the growth law $\lvert\sin z\rvert^2=\sin^2x+\sinh^2y$ .

Step 1. Expand with the addition formula, which holds for complex arguments because it follows from the exponential definitions: $\sin(x+iy)=\sin x\cos(iy)+\cos x\sin(iy)=\sin x\cosh y+i\cos x\sinh y$ , using $\cos(iy)=\cosh y$ and $\sin(iy)=i\sinh y$ .

Step 2. Insert $x=1$ , $y=1$ with $\sin1=0.84147$ , $\cos1=0.54030$ , $\cosh1=1.54308$ , $\sinh1=1.17520$ : the real part is $0.84147\times1.54308=1.29846$ and the imaginary part is $0.54030\times1.17520=0.63496$ . Hence $\sin(1+i)=\mathbf{1.29846+0.63496\,i}$ .

Step 3. Take the squared modulus: $|\sin(1+i)|^2=1.29846^2+0.63496^2=1.68600+0.40317=\mathbf{2.08917}$ , so $|\sin(1+i)|=1.44540$ . Note this exceeds $1$ — the complex sine is not bounded by unity.

Step 4. Check the growth law. From Step 1, $|\sin z|^2=\sin^2x\cosh^2y+\cos^2x\sinh^2y=\sin^2x(1+\sinh^2y)+\cos^2x\sinh^2y=\sin^2x+\sinh^2y$ . With $\sin^2 1=0.70807$ and $\sinh^2 1=1.38109$ this gives $0.70807+1.38109=2.08917$ , matching Step 3 exactly. $\sin(1+i)=1.2985+0.6350\,i$ , and $|\sin z|^2=\sin^2x+\sinh^2y$ grows without bound as $|y|\to\infty$ .

Physics of the portrait

No single person owns the complex plane, but one owns its functions. The exponential, the logarithm as an infinite family, and above all the formula $e^{i\theta}=\cos\theta+i\sin\theta$ are the legacy of Leonhard Euler, who treated the transition from real to imaginary argument as routine calculation a century before anyone made it rigorous.

Origins · 1 minLeonhard Euler and how this was discovered

Leonhard Euler

1707–1783 · the exponential and Euler's formula, founder of analytic function theory

Leonhard Euler was the most prolific mathematician in history, and the elementary complex functions are among his most consequential inventions. Born in Basel and trained under Johann Bernoulli, he worked at the academies of St Petersburg and Berlin, producing some 850 works even after going blind in 1766 — he simply dictated. His 1748 Introductio in analysin infinitorum laid out the modern treatment of functions and published the formula that bears his name, $e^{i\theta}=\cos\theta+i\sin\theta$ , obtained by boldly feeding an imaginary argument into the exponential series. From it he read off $e^{i\pi}+1=0$ , the identity often voted the most beautiful in mathematics, uniting $e$ , $i$ , $\pi$ , $1$ and $0$ in a single line. Euler defined the complex exponential, built the trigonometric functions from it, and grappled with the logarithm's multivaluedness in a famous exchange with d'Alembert — correctly insisting that $\log z$ takes infinitely many values while his rival wanted one, a dispute settled only when Riemann's surfaces made the many-valuedness geometrically inevitable. The rigor came later, from Cauchy, Riemann and Weierstrass, but the objects, the identities, and the audacious style were Euler's, and every phasor, quantum phase, and contour integral is written in his language.

History

1748

— Euler's Introductio publishes e^{iθ} = cos θ + i sin θ and the complex exponential

1749

— Euler argues in his exchange with d'Alembert that log z is infinitely many-valued

1797

— Wessel represents complex numbers geometrically, grounding the exponential map

1821

— Cauchy's Cours d'analyse puts complex functions and convergence on rigorous footing

1851

— Riemann introduces Riemann surfaces, making the multivaluedness of log and powers geometric

Going further

Texts. Stone & Goldbart, Mathematics for Physics, ch. 17 (Complex Variables) · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., ch. 11 · Hassani, Mathematical Physics, 2nd ed., Part II · Byron & Fuller, Mathematics of Classical and Quantum Physics, ch. 6 · Nakahara, Geometry, Topology and Physics, 2nd ed., §1 (Riemann-surface viewpoint) · Ahlfors, Complex Analysis, 3rd ed., ch. 2 · Needham, Visual Complex Analysis, ch. 2. Key numerical anchors for this material: the Euler series split recovers $e^{i\pi}=-1$ ; the zeros of $\sin z$ occur at $z=n\pi$ from $e^{2iz}=1$ ; $\operatorname{Log}(-1)=i\pi$ with full set $i\pi+2\pi ik$ ; $i^i=e^{-\pi/2}=0.20788$ for the principal value, with values $e^{-\pi/2-2\pi k}$ ; $\sin(1+i)=1.29846+0.63496\,i$ and $|\sin z|^2=\sin^2x+\sinh^2y$ ; $\operatorname{Log}(1+i)=\tfrac12\ln2+i\tfrac\pi4$ and $(1+i)^i=0.42883+0.15487\,i$ .

Prerequisites. Complex Numbers and the Complex Plane · Functions of a Complex Variable and Limits · Analytic Functions and the Cauchy-Riemann Equations. Leads to. Taylor Series of Analytic Functions · Laurent Series · Residues and the Residue Theorem · Zeros, Poles, and Isolated Singularities.

:::faq[How can $\sin z$ be larger than 1 when real sine never leaves $[-1,1]$ ?] Because the bound $|\sin x|\le1$ is a fact about the real axis only, and it evaporates the moment you move off it. Writing $\sin z=\sin x\cosh y+i\cos x\sinh y$ shows that the imaginary part of the argument, $y$ , enters through the hyperbolic functions $\cosh y$ and $\sinh y$ , which grow exponentially. The clean statement is $|\sin z|^2=\sin^2x+\sinh^2y$ , so as $|y|\to\infty$ the modulus grows like $\tfrac12 e^{|y|}$ without limit. This is forced by Liouville's theorem: a bounded entire function must be constant, so any non-constant entire function like $\sin z$ has to be unbounded somewhere. It is exactly this exponential growth that you must respect when closing contours in the upper or lower half-plane.

Ask the tutor: Ask me to evaluate any complex exponential, logarithm, or power step by step — pick a branch, and watch the multivaluedness turn into a clean number.

Why does making the variable complex unify so many functions?

Because they were never really separate — they only looked separate because the real line is too small to see the connections. Exponential growth ( $e^x$ ) and oscillation ( $\cos x$ , $\sin x$ ) seem unrelated on the reals, yet both are built from the same power series, and the imaginary unit is exactly the ingredient that turns one into the other. Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ makes the link explicit: sine and cosine are combinations of $e^{iz}$ and $e^{-iz}$ , hyperbolic functions are the same combinations without the $i$ , and the logarithm and powers are inverses of the exponential. Complexifying does not add structure so much as reveal what a one-dimensional domain hid.

What exactly does it mean for the logarithm to be "multivalued"?

It means $e^w=z$ has more than one solution $w$ for each nonzero $z$ — in fact infinitely many, all differing by integer multiples of $2\pi i$ . This happens because the exponential is periodic in the imaginary direction: adding $2\pi i$ to $w$ leaves $e^w$ unchanged, so you cannot recover a unique $w$ from $z$ alone. The real part $\ln|z|$ is pinned down, but the argument is only fixed modulo $2\pi$ . To get a single-valued function you pick one branch, usually the principal one with argument in $(-\pi,\pi]$ , and cut the plane so you cannot wander continuously between branches. The multivaluedness is not a defect; it is the correct description of an inverse to a periodic map.

Why must the plane be cut, and does the cut have to be the negative real axis?

The cut makes a single-valued branch possible: without it you could walk a loop around the origin and return to a different value of $\log z$ , since the argument would have increased by $2\pi$ . A cut is a barrier forbidding such loops, forcing continuity by preventing you from encircling the branch point at the origin. It need not be the negative real axis — that is a convention, chosen to keep the cut off the positive reals where most real-variable formulas live. You could equally cut along any ray from $0$ to $\infty$ ; each choice gives a different principal branch, agreeing away from the cut but jumping across its own cut. The branch point itself, $z=0$ , is intrinsic and cannot be moved.

Are the complex trig functions and the hyperbolic functions really the same thing?

Essentially yes — they are one family viewed along two perpendicular axes. The bridge identities $\cos(iz)=\cosh z$ and $\sin(iz)=i\sinh z$ say that a trig function on the imaginary axis is a hyperbolic function on the real axis, and vice versa. Both families are built from the same exponentials: $\cos z=(e^{iz}+e^{-iz})/2$ differs from $\cosh z=(e^z+e^{-z})/2$ only by where the factors of $i$ sit. Every trigonometric identity therefore has a hyperbolic twin obtained by $z\to iz$ , which is why $\cos^2z+\sin^2z=1$ maps to $\cosh^2z-\sinh^2z=1$ with its telltale sign flip. On the real axis they look like different graphs; in the complex plane the distinction is only a change of direction.

Why does

\log(z_1z_2)=\log z_1+\log z_2

fail for the principal branch?

Because the principal argument is confined to the interval $(-\pi,\pi]$ , and adding two arguments can push the sum out of that window. When it does, the principal value of $\operatorname{Arg}(z_1z_2)$ differs from $\operatorname{Arg}z_1+\operatorname{Arg}z_2$ by exactly $\pm2\pi$ , so the two sides of the identity disagree by $\pm2\pi i$ . For example, with $z_1=z_2=e^{i3\pi/4}$ each has argument $3\pi/4$ , but their product has argument $3\pi/2$ , which the principal branch reports as $-\pi/2$ — off by $2\pi$ . The identity is perfectly true as a statement about the full multivalued logarithm, where both sides are infinite sets that coincide, but it is not true value-by-value on any single branch. This is the single most common source of sign and factor errors in contour integration.

Elementary Complex Functions

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The infinitely many values of $i^i$ (confidence builder)

2 · Complex sine off the real axis (fully worked)

Physics of the portrait

Going further

Before this

Where this leads

Elementary Complex Functions

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The infinitely many values of iii^iii (confidence builder)

2 · Complex sine off the real axis (fully worked)

Physics of the portrait

Going further

Before this

Where this leads

1 · The infinitely many values of $i^i$ (confidence builder)