implicit-exponential-curve

Implicit Exponential Curve $z^y=y^z$

Summary: §519. A curve with both coordinates in exponents. The line $y=z$ is one component; the equation also has a non-trivial branch $RS$ asymptotic to the axes, parametrized by $z=(1+1/u{)}^u$, $y=(1+1/u{)}^{u+1}$, meeting the line at $z=y=e$. Drawn as figure 103.

Sources: chapter21 §519, figures103-105 (figure 103).

Last updated: 2026-05-12

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The equation (§519)

$z^y=y^z.$

The natural solution is the straight line $y=z$ (since $z^z=z^z$), but this is not the whole curve. For example $z=2,y=4$ also satisfies it: $2^4=4^2=16$.

Solution by substitution (§519)

Substitute $y=tz$:

$z^{tz}=(tz{)}^z,\text{i.e. }{z}^t=tz\text{ after extracting the zth root}.$

So $z^{t-1}=t$, giving

$z=t^{1/(t-1)},y=tz=t^{t/(t-1)}.$

Letting $t-1=1/u$ (so $t=1+1/u$) gives the cleaner parametrization:

$z={\left(1+\frac{1}{u}\right)}^u,y={\left(1+\frac{1}{u}\right)}^{u+1}.$

As $u\to \infty$: $z\to e,y\to e$ — both factors $(1+1/u{)}^u$ tend to $e$, and the extra factor $(1+1/u)\to 1$. So the parametric branch meets the diagonal $y=z$ at $(e,e)$.

The figure (§519, figure 103)

Reading figure 103: axis $AH$ horizontal, $AE$ vertical. The straight line $EAF$ (at $45°$) is the diagonal $y=z$. The branch $RS$ is the parametric curve above; it is asymptotic to lines $AG$ (horizontal) and $AH$ (vertical) — i.e., to both coordinate axes. The line $AF$ is a diameter of the branch $RS$, and the branch intersects $AF$ at $C$ where $AB=BC=e$.

So the picture has two components:

1. The straight line $EAF$ : $y=z$.
2. The hyperbolic-shaped branch $RS$ between the two axes, with axis of symmetry along $y=z$, passing through $(e,e)$.

There are also infinitely many discrete solution points — pairs of rational numbers as below.

Rational solution pairs

Take $t=3/2$ in $z=t^{1/(t-1)},y=t^{t/(t-1)}$, i.e. $u=2$. Then

$z=(3/2{)}^2=9/4,y=(3/2{)}^3=27/8.$

Check: $z^y=(9/4{)}^{27/8}$ and $y^z=(27/8{)}^{9/4}$ both equal $(3/2{)}^{27/4}$.

The full table:

$u$	$z$	$y$
1	$2$	$4$
2	$9/4$	$27/8$
3	$64/27$	$256/81$
4	$625/256$	$3125/1024$
$⋮$	$⋮$	$⋮$

General: $z=(n+1{)}^n/n^n,y=(n+1{)}^{n+1}/n^{n+1}$ for $n=1,2,3,\dots$

The square of the first equals the first raised to the power of the second:

$z^y=y^z\Rightarrow 2^4=4^2=16,(9/4{)}^{27/8}=(27/8{)}^{9/4},\dots$

Place in the chapter

Euler treats §519 as the technical climax of the §518 exponential-curve genus: the genus contains not just “one variable in an exponent” ($y=x^x$) but “both variables in exponents”, and the latter yields a nontrivial implicit curve requiring substitution tricks to even parametrize.

Figures

Figures 103–105

Related pages

• chapter-21-on-transcendental-curves
• exponential-curves — same genus, simpler equation $y=x^x$.
• logarithmic-curve — overarching genus (any equation involving $\log$).
• transcendental-curves