Proof of Estimates for the Complex Exponential and Its Continuity

Routine facts about numbers, maps and finite sums are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity); the results on conjugates and moduli cited below are read for the present complex numbers, conjugate and modulus by The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (ii), The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (iii) and The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (iv), and the results on real sequences, real series and the real exponential in the present definitions of sequences, convergence, limits and series by The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i), The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (ii), The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (iii) and The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (v).

Partial sums. Fix $h\in\mathbb{C}$. Let

$$S_{n}=\sum_{k=1}^{n}\frac{h^{k-1}}{(k-1)!},\qquad s_{n}=\sum_{k=1}^{n}\frac{|h|^{k-1}}{(k-1)!}\qquad(n\in\mathbb{N}).$$

The $S_{n}$ are the partial sums of the series defining $\exp h$ (The Complex Exponential Function (i)), so $S_{n}\to\exp h$ (Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii)). The $s_{n}$ are the partial sums of the real exponential series at $|h|$; this series converges (The Exponential Series Converges Absolutely, and Its Sum Is a Map on ℝ (ii)), and its sum is $\exp_{\mathbb{R}}(|h|)$ (The Real Exponential Function (i), The Real Exponential Function (ii)). Its terms are nonnegative, so its partial sums lie below its sum (Series of Real Numbers with Nonnegative Terms, Comparison, Absolute Convergence and the Geometric Series (i)):

$$0\le s_{n}\le\exp_{\mathbb{R}}(|h|)\qquad(n\in\mathbb{N}).$$

The first remainder, before the limit. Let $n\ge2$. Splitting off the term $h^{0}/0!=1$ of index $k=1$, we have $S_{n}-1=\sum_{k=2}^{n}h^{k-1}/(k-1)!$. Here $|h^{k-1}|=|h|^{k-1}$, by induction on $k$ from $|1|=1$ (The Modulus of a Complex Number (ii)) and $|zw|=|z|\,|w|$ (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (iv)), and $|1/(k-1)!|=1/(k-1)!$, since the modulus of the positive real number $(k-1)!$ is its absolute value (The Modulus of a Complex Number (ii)) and the modulus of a reciprocal is the reciprocal of the modulus (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (v)); so, by multiplicativity again, $|h^{k-1}/(k-1)!|=|h|^{k-1}/(k-1)!$. Bounding the modulus of the finite sum by the sum of the moduli of its terms (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (viii)), and then substituting $k=j+1$, we get

$$|S_{n}-1|\le\sum_{k=2}^{n}\frac{|h|^{k-1}}{(k-1)!}=|h|\sum_{j=1}^{n-1}\frac{|h|^{j-1}}{j!}\le|h|\,s_{n-1}\le|h|\,\exp_{\mathbb{R}}(|h|).$$

The middle inequality compares term by term, using $0<(j-1)!\le j!$ (Natural Numbers and Factorials Read in the Reals: the Order Carries Over, One Is Least, and Factorials Are Positive, Recursive and Nondecreasing (iii) with $k=j-1$).

The second remainder, before the limit. Let $n\ge3$. In the same way, now also splitting off the term $h^{1}/1!=h$ of index $k=2$ and substituting $k=j+2$,

$$|S_{n}-1-h|\le\sum_{k=3}^{n}\frac{|h|^{k-1}}{(k-1)!}=|h|^{2}\sum_{j=1}^{n-2}\frac{|h|^{j-1}}{(j+1)!}\le|h|^{2}\,s_{n-2}\le|h|^{2}\,\exp_{\mathbb{R}}(|h|),$$

where now $0<(j-1)!\le j!\le(j+1)!$ (Natural Numbers and Factorials Read in the Reals: the Order Carries Over, One Is Least, and Factorials Are Positive, Recursive and Nondecreasing (iii) with $k=j-1$ and with $k=j$).

Passing to the limit. Since $S_{n}\to\exp h$, the sequences $S_{n}-1$ and $S_{n}-1-h$ converge to $\exp h-1$ and $\exp h-1-h$ (Limits of Complex Sequences: Uniqueness, Real and Imaginary Parts, Arithmetic, Real Sequences, Conjugates, Moduli, the Cauchy Criterion and the Modulus Metric (iii)), and so their moduli converge to the moduli of these limits (Limits of Complex Sequences: Uniqueness, Real and Imaginary Parts, Arithmetic, Real Sequences, Conjugates, Moduli, the Cauchy Criterion and the Modulus Metric (v)). The bounds of the two previous paragraphs hold for all $n\ge2$ and all $n\ge3$ respectively, and their right-hand sides do not depend on $n$; so, comparing with these constants (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (i), Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (x)), they persist in the limit:

$$|\exp h-1|\le|h|\,\exp_{\mathbb{R}}(|h|),\qquad|\exp h-1-h|\le|h|^{2}\,\exp_{\mathbb{R}}(|h|).$$

As $h$ was arbitrary, this proves (i).

Continuity. Fix $z\in\mathbb{C}$. For $w\in\mathbb{C}$ put $h=w-z$. The functional equation (The Complex Exponential: the Functional Equation, Conjugates, Real Arguments and the Modulus (ii)) gives $\exp w=\exp z\cdot\exp h$, so $\exp w-\exp z=\exp z\,(\exp h-1)$, and by multiplicativity of the modulus (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (iv)) and (i),

$$d_{\mathbb{C}}(\exp w,\exp z)=|\exp w-\exp z|\le|\exp z|\,|h|\,\exp_{\mathbb{R}}(|h|).$$

Two constants in this bound are positive: $|\exp z|>0$, since $\exp z\neq0$ (The Complex Exponential: the Functional Equation, Conjugates, Real Arguments and the Modulus (iii)) and only $0$ has modulus $0$ (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (iii)); and $e=\exp_{\mathbb{R}}(1)>0$ (Basic Properties of the Exponential Function (iii)). Now let a real $\varepsilon>0$ be given, and choose, depending on $\varepsilon$ and $z$,

$$\delta=\min\Big\{1,\ \frac{\varepsilon}{e\,|\exp z|}\Big\}>0.$$

Let $w\in\mathbb{C}$ with $d_{\mathbb{C}}(z,w)<\delta$; here $d_{\mathbb{C}}(z,w)=|z-w|=|-h|=|h|$ (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (iii)), so $|h|<\delta$. Then $|h|<1$, so $\exp_{\mathbb{R}}(|h|)<e$ because $\exp_{\mathbb{R}}$ is increasing (Basic Properties of the Exponential Function (ix)), and

$$d_{\mathbb{C}}(\exp w,\exp z)\le|\exp z|\,e\,|h|<|\exp z|\,e\,\delta\le\varepsilon.$$

So $\exp$ is continuous at $z$; as $z\in\mathbb{C}$ was arbitrary, $\exp$ is continuous (Continuous Map Between Metric Spaces (ii)). This proves (ii).

Difference quotients. Let $h\in\mathbb{C}$ with $h\neq0$. By the functional equation (The Complex Exponential: the Functional Equation, Conjugates, Real Arguments and the Modulus (ii)), $\exp(z+h)-\exp z=\exp z\,(\exp h-1)$, and therefore

$$\frac{\exp(z+h)-\exp z}{h}-\exp z=\exp z\cdot\frac{\exp h-1-h}{h}.$$

By multiplicativity of the modulus and the modulus of a reciprocal (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (iv), Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (v)), and then (i), the modulus of the right-hand side is

$$|\exp z|\,\frac{|\exp h-1-h|}{|h|}\le|\exp z|\,\frac{|h|^{2}\exp_{\mathbb{R}}(|h|)}{|h|}=|\exp z|\,|h|\,\exp_{\mathbb{R}}(|h|).$$

This proves (iii).