Proof of Jordan's lemma by Jordan's inequality

Builds on Contour integral Complex exponential Product of exponentials Euler's formula Pythagorean trigonometric identity Modulus is multiplicative Monotonicity of the integral Integration by substitution Mean value theorem A function with nonnegative derivative is increasing The exponential function is strictly increasing

Proof

Let $a$ , $R$ , $C_R$ , $g$ , $f$ and $M_R$ be as in the statement. By the definition of the contour integral, $\int_{C_R} f(z) \, dz = \int_0^\pi e^{iaRe^{i\theta}} \, g(Re^{i\theta}) \, iRe^{i\theta} \, d\theta.$
For real $t$ , $\abs{e^{it}} = \abs{\cos t + i \sin t} = 1$ , by Euler's formula and the Pythagorean identity.
We have $iaRe^{i\theta} = -aR\sin\theta + iaR\cos\theta$ . By the product of exponentials, step 2 and the multiplicativity of the modulus, $\abs{e^{iaRe^{i\theta}}} = e^{-aR\sin\theta}$ .
So the modulus of the integrand in step 1 is at most $e^{-aR\sin\theta} M_R R$ , since $\abs{iRe^{i\theta}} = R$ .
For a continuous complex function $k$ on $[0, \pi]$ , write $\int_0^\pi k = \abs{\int_0^\pi k} \, e^{i\alpha}$ with $\alpha$ real. Then $\abs{\int_0^\pi k} = \int_0^\pi \operatorname{Re}(e^{-i\alpha} k) \le \int_0^\pi \abs{k}$ , by monotonicity of the integral.
By steps 4 and 5, $\abs{\int_{C_R} f(z) \, dz} \le R M_R \int_0^\pi e^{-aR\sin\theta} \, d\theta$ .
The substitution $\theta \mapsto \pi - \theta$ and $\sin(\pi - \theta) = \sin\theta$ give $\int_{\pi/2}^{\pi} e^{-aR\sin\theta} \, d\theta = \int_0^{\pi/2} e^{-aR\sin\theta} \, d\theta$ . So the integral in step 6 is $2 \int_0^{\pi/2} e^{-aR\sin\theta} \, d\theta$ .
Jordan's inequality: $\sin\theta \ge 2\theta/\pi$ for $\theta \in [0, \pi/2]$ . Let $h(\theta) = \sin\theta - 2\theta/\pi$ . Then $h(0) = h(\pi/2) = 0$ , and $h'(\theta) = \cos\theta - 2/\pi$ is strictly decreasing on $[0, \pi/2]$ .
By the mean value theorem there is $\xi \in (0, \pi/2)$ with $h'(\xi) = 0$ . So $h' \ge 0$ on $[0, \xi]$ and $h' \le 0$ on $[\xi, \pi/2]$ .
By nonnegative derivative implies increasing, applied to $h$ on $[0, \xi]$ and to $-h$ on $[\xi, \pi/2]$ , we get $h \ge h(0) = 0$ and $h \ge h(\pi/2) = 0$ . This proves Jordan's inequality.
Since $a R > 0$ and the exponential function is increasing, step 10 gives $e^{-aR\sin\theta} \le e^{-2aR\theta/\pi}$ on $[0, \pi/2]$ .
By monotonicity of the integral, $\int_0^{\pi/2} e^{-aR\sin\theta} \, d\theta \le \int_0^{\pi/2} e^{-2aR\theta/\pi} \, d\theta = \frac{\pi}{2aR} \bigl( 1 - e^{-aR} \bigr) \le \frac{\pi}{2aR}$ .
Combining steps 6, 7 and 12, $\abs{\int_{C_R} f(z) \, dz} \le R M_R \cdot 2 \cdot \frac{\pi}{2aR} = \frac{\pi}{a} M_R$ .

Idea

On the upper semicircle, $\abs{e^{iaz}} = e^{-aR\sin\theta}$ is very small except near the two ends, where $\sin\theta$ is close to $0$ . Jordan's inequality $\sin\theta \ge 2\theta/\pi$ bounds this factor by an exponential that integrates to about $\pi/(2aR)$ . This factor $1/R$ cancels the length $\pi R$ of the semicircle.

Reading path

To understand this page from the ground up, read these 325 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the Cauchy–Hadamard theorem, Proof of the derivative of cosine by term-by-term differentiation, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the formula for the derivative of a polynomial, Proof of the commutativity of addition by induction, Proof of term-by-term differentiation, Proof of the epsilon-delta form of continuity, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof of the reverse triangle inequality, Proof that the exponential series has infinite radius of convergence, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a function with nonnegative derivative is increasing, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that the exponential function is positive, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that Lipschitz functions are uniformly continuous, Proof that the integral of a continuous function is an antiderivative, Proof of strong induction from ordinary induction, Proof that power series are differentiable inside the radius of convergence, Proof of the limit of a product of sequences, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of Pascal's rule from the recursive definition, Proof that i squared is minus one, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof that the exponential function is strictly increasing, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the first fundamental theorem of calculus, Proof that the modulus is multiplicative, Proof of the comparison test, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the linearity of the integral, Proof of the factorial formula for binomial coefficients by induction, Proof that a function with positive derivative is strictly increasing, Proof of integration by substitution, Proof of the product of exponentials by the Cauchy product, Proof of the second fundamental theorem of calculus, Proof that the complex numbers form a field, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof of the epsilon-delta form of the limit of a function, Proof of the derivative of sine by term-by-term differentiation, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a composition of continuous functions is continuous, Proof of the exponential of a sum by the Cauchy product, Proof of the Pythagorean trigonometric identity by differentiation, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the exponential function is its own derivative, by term-by-term differentiation, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that a function with zero derivative on an open interval is constant, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the ratio formula for the radius of convergence, Proof of Euler's formula by power series, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof that sums and products of continuous functions are continuous, Proof that continuous functions are Riemann integrable, Proof of the constant multiple rule.

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Proof of the intermediate value theorem by the supremum proof
Proof that the reals are Archimedean from the completeness axiom proof
Proof of the reverse triangle inequality proof
Proof of the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Proof of the uniqueness of limits proof
Proof that limits preserve non-strict inequalities proof
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Proof that differentiable functions are continuous proof
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Proof that convergent sequences are bounded proof
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Proof of the term test proof
Proof of the monotone convergence theorem for sequences proof
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Differentiable functions are continuous statement
Limit of a sum of sequences statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
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Convergent sequences are bounded statement
Proof of the monotonicity of the integral proof
Proof of the ratio formula for the radius of convergence proof
Proof that Darboux and Riemann integrals agree proof
Subsequences of a convergent sequence converge to the same limit statement
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
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A complex number times its conjugate statement
Proof of the limit of a product of sequences proof
There is a natural number n with 1/n below any positive epsilon statement
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
Term test statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
The sequence r^n tends to zero for |r| < 1 example
Proof that a decreasing sequence bounded below converges proof
Proof that the modulus is multiplicative proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Limit of a product of sequences statement
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A series of nonnegative terms converges iff its partial sums are bounded statement
Proof of Riemann's criterion for integrability proof
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Continuity is equivalent to sequential continuity statement
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Proof of the root test proof
Sequential characterization of limits statement
Proof of the ratio test proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
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Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
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Proof that sums and products of continuous functions are continuous proof
Comparison test statement
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Bolzano–Weierstrass theorem statement
Exponentials dominate powers in sequences statement
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Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Ratio test statement
Additivity of the integral over intervals statement
Root test statement
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Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Product of exponentials statement
Proof of the constant multiple rule proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Proof of the power rule by induction proof
Boundedness theorem statement
Proof of the first fundamental theorem of calculus proof
Ratio formula for the radius of convergence statement
Cauchy–Hadamard theorem statement
Cauchy criterion for sequences statement
Constant multiple rule statement
Power rule statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Extreme value theorem statement
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Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
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Proof that continuous functions are Riemann integrable proof
Proof that absolutely convergent series converge proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
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Rolle's theorem statement
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Proof of the Weierstrass M-test proof
Mean value theorem for integrals statement
Proof of the mean value theorem from Rolle's theorem proof
Weierstrass M-test statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Proof that a function with zero derivative on an open interval is constant proof
Proof that a function with nonnegative derivative is increasing proof
Proof that the integral of a continuous function is an antiderivative proof
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
Proof that a function with positive derivative is strictly increasing proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
Proof of the second fundamental theorem of calculus proof
A function with zero derivative is constant statement
A function with nonnegative derivative is increasing statement
The integral of a continuous function is an antiderivative statement
Uniform convergence of derivatives allows term-by-term differentiation statement
Second fundamental theorem of calculus statement
A function with positive derivative is strictly increasing statement
Proof that the exponential series has infinite radius of convergence proof
Proof of integration by substitution proof
Proof of term-by-term differentiation proof
The exponential series has infinite radius of convergence statement
Integration by substitution statement
Term-by-term differentiation statement
Exponential function definition
Proof of the exponential of a sum by the Cauchy product proof
Proof that power series are differentiable inside the radius of convergence proof
Exponential of a sum statement
Power series are differentiable inside the radius of convergence statement
Proof that the exponential function is positive proof
Proof that the exponential function is its own derivative, by term-by-term differentiation proof
Proof of the derivative of cosine by term-by-term differentiation proof
Proof of the derivative of sine by term-by-term differentiation proof
The exponential function is positive statement
The exponential function is its own derivative statement
Derivative of cosine statement
Derivative of sine statement
Proof that the exponential function is strictly increasing proof
Proof of the Pythagorean trigonometric identity by differentiation proof
The exponential function is strictly increasing statement
Pythagorean trigonometric identity statement
Proof of Jordan's lemma by Jordan's inequality proof

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Jordan's lemma

Tag	`B8NK5QQCQS`
URI	`m2://B8NK5QQCQS`
Kind	proof
Writing	W0 stub
MSC	30E20
Proves	Jordan's lemma
Uses	Contour integral, Complex exponential, Product of exponentials, Euler's formula, Pythagorean trigonometric identity, Modulus is multiplicative, Monotonicity of the integral, Integration by substitution, Mean value theorem, A function with nonnegative derivative is increasing, The exponential function is strictly increasing
Changed	4 November 2026

Proof of Jordan's lemma by Jordan's inequality

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