Integral transforms, operational calculus Proof

Proof of the formula for the Laplace transform of a derivative by integration by parts

Builds on Laplace transform Product of exponentials Euler's formula for a real angle, both signs Derivative of the complex exponential Product rule Second fundamental theorem of calculus Continuous functions are Riemann integrable Integral of the exponential function Comparison test for improper integrals Integration by substitution

Proof

We integrate by parts on $[0, T]$ and let $T \to \infty$ .

Fix $s \in \mathbb{C}$ with $\sigma = \operatorname{Re} s > a$ , and write $s = \sigma + i \omega$ with $\omega$ real. Let $E(t) = e^{-st}$ for $t \ge 0$ . Note that $M \ge \abs{f(0)} \ge 0$ .
By the product of exponentials, $E(t) = e^{-\sigma t} e^{-i \omega t}$ . By Euler's formula, $\abs{e^{-i\omega t}} = \abs{\cos(\omega t) - i \sin(\omega t)} = 1$ . So $\abs{E(t)} = e^{-\sigma t}$ .
For real $h \ne 0$ , $\dfrac{E(t + h) - E(t)}{h} = E(t)\, \dfrac{e^{-sh} - 1}{h}$ , by the product of exponentials. If $s \ne 0$ , this quotient tends to $-s E(t)$ as $h \to 0$ , by the derivative of the complex exponential at $0$ . If $s = 0$ , it is $0$ . So $E'(t) = -s E(t)$ .
Let $g(t) = M e^{-(\sigma - a) t}$ . By step 2, $\abs{f(t) E(t)} \le M e^{at} e^{-\sigma t} = g(t)$ for $t \ge 0$ .
By the integral of the exponential function and integration by substitution with $x = -(\sigma - a) t$ , $\int_0^T g(t)\, dt = \frac{M}{\sigma - a}\left( 1 - e^{-(\sigma - a) T} \right)$ . This tends to $\frac{M}{\sigma - a}$ as $T \to \infty$ , since $\sigma - a > 0$ .
Let $p$ be the real part of $t \mapsto f(t) E(t)$ . It is continuous, and $0 \le p(t) + g(t) \le 2 g(t)$ by step 4.
By the comparison test for improper integrals and step 5, $\int_0^\infty (p + g)$ converges. So $\int_0^T p = \int_0^T (p + g) - \int_0^T g$ has a limit as $T \to \infty$ .
The same argument applies to the imaginary part. So the limit $F(s) = \lim_{T \to \infty} \int_0^T f(t) E(t)\, dt$ exists.
The function $f'$ is continuous, so it is Riemann integrable on each $[0, T]$ . Thus its Laplace transform is defined as a limit, where the limit exists.
By the product rule, applied to real and imaginary parts, and step 3, $(f E)' = f' E - s f E$ . This derivative is continuous on $[0, \infty)$ .
By the second fundamental theorem of calculus, applied to the real and imaginary parts of $f E$ on $[0, T]$ , we get $f(T) E(T) - f(0) = \int_0^T f'(t) E(t)\, dt - s \int_0^T f(t) E(t)\, dt$ .
So $\int_0^T f'(t) e^{-st}\, dt = f(T) e^{-sT} - f(0) + s \int_0^T f(t) e^{-st}\, dt$ for each $T > 0$ .
By step 4, $\abs{f(T) e^{-sT}} \le M e^{-(\sigma - a) T}$ , and this tends to $0$ as $T \to \infty$ .
By steps 8 and 13, the right side of step 12 tends to $0 - f(0) + s F(s)$ as $T \to \infty$ .
Hence the Laplace transform of $f'$ converges at $s$ , and $\mathcal{L}\{f'\}(s) = s F(s) - f(0)$ .

Idea

Integration by parts moves the derivative from $f$ onto $e^{-st}$ , and differentiating $e^{-st}$ only multiplies it by $-s$ . The boundary term at $0$ gives $-f(0)$ . The boundary term at $T$ vanishes in the limit because $f$ grows at most like $e^{at}$ , while $e^{-st}$ decays like $e^{-\operatorname{Re}(s) t}$ .

Reading path

To understand this page from the ground up, read these 351 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 that the complex exponential is its own derivative, 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 diameter characterisation of Cauchy sequences, 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 triangle inequality for norms, 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 integral of the exponential function, 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 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 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 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 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 the existence of the infimum by negation, 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 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 that one-sided limits of a monotone function exist, Proof of the epsilon-delta form of the limit of a function, 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 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 absolutely convergent series in a Banach space converge, Proof that power series are holomorphic in the disc of convergence, Proof of the monotonicity of the integral, Proof of the comparison test for improper integrals, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the ratio formula for the radius of convergence, 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.

Binomial coefficient definition
Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Element definition
Extended real number line object
Factorial definition
Function concept
Group concept
Imaginary unit object
Integers object
Limit point of a set definition
Lipschitz continuity definition
Local maximum definition
Mathematical induction concept
Natural numbers object
Norm concept
Ordered field definition
Partition of an interval definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Vector space definition
Proof of Pascal's rule from the recursive definition proof
Complex exponential definition
Power series in the complex plane definition
Real and imaginary parts definition
Set definition
Laplace transform definition
Ring definition
Ring of integers statement
Limit of a function definition
Local minimum definition
Archimedean ordered field definition
Successor function definition
Mesh of a partition definition
Refinement of a partition definition
Riemann sum definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Norm definition
Pascal's rule statement
Complex sine and cosine definition
Complex conjugate definition
Disc of convergence definition
Empty set object
Family of sets definition
Ordered pair definition
Commutative ring definition
Unit of a ring definition
Continuity at a point definition
Derivative definition
Intersection definition
Limit at infinity definition
One-sided limit definition
Union definition
Addition of natural numbers definition
Riemann integrable function definition
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Biconditional definition
Truth table concept
Modus ponens concept
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Sequence tending to infinity definition
Series definition
Completeness axiom statement
Euler's formula for a real angle, both signs statement
Proof of the factorial formula for binomial coefficients by induction proof
Proof of the equality of complex numbers proof
Cartesian product definition
Field definition
Continuous function definition
Antiderivative definition
Differentiable function definition
One-sided derivative definition
Subset definition
Archimedean property statement
Proof of the commutativity of addition by induction proof
Union of a family of sets definition
Order on the natural numbers definition
Riemann integral definition
Propositional logic concept
The rationals form an ordered field statement
Upper and lower Darboux sums definition
Mathematical proof concept
Uniqueness quantifier definition
Limit superior definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Pointwise convergence definition
Sequential continuity definition
The sequence r^n tends to infinity for r > 1 example
Cauchy product definition
Partial sum definition
Power series definition
Equality of complex numbers statement
Factorial formula for binomial coefficients statement
Binary relation definition
Proof that a composition of continuous functions is continuous proof
Derivative function definition
Topological space definition
Order on the reals definition
Improper integral definition
Commutativity of addition statement
Definite integral definition
Proof of strong induction from ordinary induction proof
Upper and lower Darboux integrals definition
Tautology definition
Uniform convergence definition
Sum of a finite geometric series statement
Convergent series definition
Radius of convergence definition
Antisymmetric relation definition
Proof of the binomial theorem by induction proof
Reflexive relation definition
Transitive relation definition
Higher derivative definition
Interval definition
Proof that the complex numbers form a field proof
Function definition
Strong induction statement
Uniform convergence on a set definition
Logical equivalence definition
Sum of a series definition
Series of functions definition
Absolute convergence definition
Binomial theorem statement
Preorder definition
Continuously differentiable function definition
Proof of the epsilon-delta form of the limit of a function proof
Proof of the epsilon-delta form of continuity proof
The complex numbers form a field statement
Metric space definition
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Monotone function definition
Epsilon-delta definition of a limit statement
Convergence of a sequence in a metric space definition
Open ball definition
Epsilon-delta definition of continuity statement
Well-ordering principle statement
Diameter of a set definition
Double negation law statement
Total order definition
Composition of continuous functions is continuous statement
Metric topology definition
Uniform continuity definition
Proof of the diameter characterisation of Cauchy sequences proof
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Uniform continuity on a set definition
Open set definition
Epsilon characterisation of Cauchy sequences statement
Proof that Lipschitz functions are uniformly continuous proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Equivalent metrics definition
Complete metric space definition
Lipschitz functions are uniformly continuous statement
The reals form an ordered field statement
Equivalent norms definition
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the existence of square roots of nonnegative reals proof
Proof of the existence of the infimum by negation proof
Banach space definition
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Existence of the infimum statement
Triangle inequality statement
Absolute value is multiplicative statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof that one-sided limits of a monotone function exist proof
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
Modulus of a complex number definition
Proof of the squeeze theorem for sequences proof
Proof of the uniqueness of limits proof
Proof that limits preserve non-strict inequalities proof
Intermediate value theorem statement
Proof that differentiable functions are continuous proof
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
The reals are Archimedean statement
The sequence 1/n tends to zero example
Proof of the term test proof
Proof that Cauchy sequences are bounded proof
Proof of the monotone convergence theorem for sequences proof
Reverse triangle inequality statement
Complex differentiability definition
Proof that a complex number times its conjugate is the square of its modulus proof
Differentiable functions are continuous statement
Proof of the triangle inequality for norms proof
Linearity of the integral statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Limit of a sum of sequences statement
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Proof of the monotonicity of the integral proof
Proof of the ratio formula for the radius of convergence proof
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
Squeeze theorem statement
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Holomorphic function definition
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
One-sided limits of a monotone function exist statement
Darboux and Riemann integrals agree statement
Triangle inequality for norms statement
Term test statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Proof that the modulus is multiplicative proof
Proof that absolutely convergent series in a Banach space converge proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Decreasing sequence bounded below converges statement
Limit of a product of sequences statement
Proof of Riemann's criterion for integrability proof
Geometric series example
A series of nonnegative terms converges iff its partial sums are bounded statement
Absolutely convergent series in a Banach space converge statement
Modulus is multiplicative statement
Continuity is equivalent to sequential continuity statement
Riemann's criterion for integrability statement
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Sequential characterization of limits statement
Proof of the root test proof
Proof of the Cauchy–Hadamard theorem proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the ratio test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the comparison test proof
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the product rule proof
Proof of the linearity of differentiation proof
Proof that sums and products of continuous functions are continuous proof
Comparison test statement
Cauchy product of absolutely convergent series statement
Proof of the additivity of the integral over intervals proof
Bolzano–Weierstrass theorem statement
Exponentials dominate powers in sequences statement
Chain rule statement
Product of continuous functions is continuous statement
Product rule statement
Proof of the product of exponentials by the Cauchy product proof
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Linearity of differentiation statement
Additivity of the integral over intervals statement
Ratio test statement
Root test statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Product of exponentials statement
Continuous functions on a closed bounded interval are uniformly continuous statement
Proof that power series are holomorphic in the disc of convergence proof
Proof of the constant multiple rule proof
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
Proof of the comparison test for improper integrals proof
Cauchy–Hadamard theorem statement
Cauchy criterion for sequences statement
Power series are holomorphic in the disc of convergence statement
Constant multiple rule statement
Power rule statement
Comparison test for improper integrals 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
Proof of the formula for the derivative of a polynomial proof
Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
Derivative of a polynomial statement
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
Absolutely convergent series converge statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of the mean value theorem for integrals proof
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 the integral of a continuous function is an antiderivative proof
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
Proof of the second fundamental theorem of calculus proof
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
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
Proof that the complex exponential is its own derivative proof
Term-by-term differentiation statement
Exponential function definition
Derivative of the complex exponential statement
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 its own derivative, by term-by-term differentiation proof
The exponential function is its own derivative statement
Proof of the integral of the exponential function proof
Integral of the exponential function statement
Proof of the formula for the Laplace transform of a derivative by integration by parts proof

Where this leads

5 pages build on this one.

Tag	`P6G7J7GK8N`
URI	`m2://P6G7J7GK8N`
Kind	proof
Writing	W0 stub
MSC	44A10
Proves	Laplace transform of a derivative
Uses	Laplace transform, Product of exponentials, Euler's formula for a real angle, both signs, Derivative of the complex exponential, Product rule, Second fundamental theorem of calculus, Continuous functions are Riemann integrable, Integral of the exponential function, Comparison test for improper integrals, Integration by substitution
Changed	4 November 2026

Proof of the formula for the Laplace transform of a derivative by integration by parts

Proof

Idea

Reading path

Where this leads

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