EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 3, 2022, 1211-1216 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the relationship between the integral of derivative and the derivative of the integral Giljun Han1, Hwajoon Kim2∗ 1Departmet of Mathematics Education, Dankook University, Yongin-si, Gyeonggi-do, S. Korea 2 IT Engineering, Kyungdong University, Yangju, Gyeonggi-do, S. Korea Abstract. We would like to study the relationship between the integral of the derivative and the derivative of the integral. This is the study of whether these two are the same or not. Also, this study began with the question of Kreyszig’s assumptions presented in [8]. The research method used two tools: the Riemann integral and its generalized concept, the Lebesgue integral. The obtained research results give some answer to this question. 2020 Mathematics Subject Classifications: 35-02 Key Words and Phrases: Leibniz integral rule, interchange of integral and derivative, monotone convergence theorem 1. Introduction Among the various methods to find the solution of PDEs, there is a method using integral transform [3–9]. There is one problem in this solution process. That is, the solution is made under the assumption that the order of differentiation and integration can be changed. This problem appears frequently in the calculation process of £(uxx) in PDEs. This point was mentioned in [8], and it seems that Kreyszig was also aware of it(Sec. 12.11 of [8]). Is it possible to interchange the integral and derivative in solving PDEs by integral transform? This study intends to start with this question. Since the solution by integral transform is almost similar, we would just like to approach by the Laplace transform. Consider a function w(x, t) = xn + tn, where n is a positive integer. By simple calcu- lation, we get ∂n ∂xn ∫ w(x, t) dt = ∫ ∂n ∂xn w(x, t) dt ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4424 Email addresses: gilhan@dankook.ac.kr (Gj. Han), cellmath@gmail.com (hj. Kim) https://www.ejpam.com 1211 © 2022 EJPAM All rights reserved. Gj. Han, Hj. Kim / Eur. J. Pure Appl. Math, 15 (3) (2022), 1211-1216 1212 under the assumption that the difference as much as constant is accepted. On one hand, as a counter example, consider the following example: Let y be the positive integers, dy be the counting measure, and f(x, y) = { x ( 1 y + 1 < x < 1 y ) 0 elsewhere . It is well-known that the integral of the derivative of f is 0 at x = 0 and the derivative of the integral is 1 at x = 0. It is necessary to find out more about to what extent equality holds and where equality does not hold. In the next section, consider a further approach to this topic. 2. On the relationship between the integral of derivative and the derivative of the integral From the point of view of the Riemann integral, this interchange is closely related to the following Leibniz integral rule. From the point of view of the Lebesgue integral, this interchange relates to monotone convergence theorem, Beppo Levi’s theorem and Lebesgue dominated convergence theorem. Therefore, let us take a look at these theorems first as lemmas. Lemma 1. (Leibniz integral rule) If f and fx are continuous functions, then d dx ( ∫ b(x) a(x) f(x, t) dt) = f(x, b(x)) · d dx b(x)− f(x, a(x)) · d dx a(x) + ∫ b(x) a(x) ∂ ∂x f(x, t) dt is valid for a(x), b(x), a′(x) and b′(x) are all continuous. If a(x) and b(x) are constants, we can easily see that d dx ( ∫ b a f(x, t) dt) = ∫ b a ∂ ∂x f(x, t) dt. Lemma 2. (Monotone convergence theorem(MCT) [1, 2]). Let M+ be the collection of all non-negative measurable functions and let µ be a measure. If (fn) is a monotone increasing sequence of functions in M+ which converges to f , then∫ f dµ = lim ∫ fn dµ. In other words, this theorem can be expressed as∫ f dµ = ∫ lim fn dµ = lim ∫ fn dµ. Gj. Han, Hj. Kim / Eur. J. Pure Appl. Math, 15 (3) (2022), 1211-1216 1213 Putting fn = g1 + · · ·+ gn and applying the MCT, then∫ ∞∑ n=1 gn dµ = ∫ lim m∑ n=1 gn dµ = lim m∑ n=1 ∫ gn dµ = ∞∑ n=1 ∫ gn dµ for gn is measurable. This is the famous Beppo Levi’s theorem. Lemma 3. (Beppo Levi’s theorem[2]). Let (gn) be a sequence in M+, then∫ ∞∑ n=1 gn dµ = ∞∑ n=1 ∫ gn dµ. Lemma 4. (Lebesgue dominated convergence theorem(LDCT) [1, 2]). Let (fn) be a se- quence of integrable functions which converges almost everywhere to a real-valued measur- able function f . If there exists an integrable function g such that |fn| ≤ g for all n, then f is integrable and ∫ f dµ = lim ∫ fndµ. As an intuitive approach, consider whether d dx ∫ b a f(x, t) dt = ∫ b a fx(x, t) dt or not. Putting g(x) = ∫ b a f(x, t) dt, we have g′(x) = lim h→0 g(x+ h)− g(x) h = lim h→0 ∫ b a f(x+ h, t) dt− ∫ b a f(x, t) dt h = lim h→0 ∫ b a (f(x+ h, t)− f(x, t)) h dt = lim h→0 ∫ b a f(x+ h, t)− f(x, t) h dt = ∫ b a fx(x, t) dt by LDCT and mean value theorem. Therefore, d dx ∫ b a f(x, t) dt = ∫ b a fx(x, t) dt, where fx is the partial derivative with respect to x. In this regard, the following lemma can be constructed: Gj. Han, Hj. Kim / Eur. J. Pure Appl. Math, 15 (3) (2022), 1211-1216 1214 Lemma 5. Let X be an open subset of R, and ω be a measure space. If f : X×ω → R sat- isfies the following conditions: (i)f is Lebesgue-integrable, (ii)fx exists almost everywhere, and (iii) there is an integrable function g such that |fx| ≤ g, then d dx ∫ ω f = ∫ ω fx. Proof. The proof follows from the LDCT and the mean value theorem. Regarding this problem, consider three places in Kreyszig’s Advanced Engineering Mathematics [8]. One is shown in the proof of theorem 1 of section 11.1, and the other two are shown in the solution of example 1 of section 12.12. Theorem 1. (1)(Interchange of infinite series and integral)∫ π −π ∞∑ n=1 (ancosnx+ bnsinnx) dx = ∞∑ n=1 (an ∫ π −π cosnx dx+ bn ∫ π −π sinnx dx). (2) (Interchange of limit and integral) Let w be the displacement of an elastic string such that w(0, t) = f(t) = { sin t (0 ≤ t ≤ 2π) 0 otherwise . Then lim x→∞ ∫ ∞ 0 e−stw(x, t) dt = ∫ ∞ 0 e−st lim x→∞ w(x, t) dt is valid for lim x→∞ w(x, t) = 0. (3) (Interchange of derivative and integral) In the function of (2),∫ ∞ 0 e−st∂ 2w ∂x2 dt = ∂2 ∂x2 ∫ ∞ 0 e−stw dt is established. Proof. (1) It suffices to show that the assumptions of Beppo Levi’s theorem are satis- fied. Since ∞∑ n=1 (ancosnx+ bnsinnx) is nonnegative valued on (−π, π) and measurable, the proof is complete. REFERENCES 1215 (2) Let us check whether the assumption of LDCT is satisfied. Note that |w(x, t)| ≤ 1 for all t and lim x→∞ w(x, t) = 0 for t ≥ 0. Therefore, the result of (2) is naturally derived. (3) From the point of view of the Riemann integral, e−st and −se−st are continuous, so the result is obtained by the Leibniz integral rule. Next, let us check it in terms of the more general Lebesgue integral. Since e−st is Lebesgue integrable on [0,∞), −se−st exists, and for some constants M and k it satisfies | − se−st| ≤ Mekt, by lemma 5, the proof is complete. Based on the above results, Kreyszig’s assumption (p 601 of [8]) can be considered unnecessary. The reason is that e−stw and its derivative are continuous. Conflict of interest. The authors declare no conflicts of interest. Acknowledgements The corresponding author (Hj. Kim) acknowledges the support of Kyungdong Univer- sity Research Fund, 2022. References [1] R. G. Bartle. The elements of integration and Lebesgue measure. John Willy Sons, Inc., New York, 1995. [2] D. L. Cohn. Measure theory. Birkh¨auser, Boston, 1980. [3] Y. H. Geum, A.K. Rathie, and Hj. Kim. Matrix Expression of Convolution and Its Generalized Continuous Form. Symmetry, 2020(12):1791, 2020. [4] S. Jirakulchaiwong, K. Nonlaopon, J. Tariboon, S. K. Ntouyas, and Hj. Kim. On (p,q)-Analogues of Laplace-Typed Integral Transforms and Applications. Symmetry, 2021(13):631, 2021. [5] Arjun K, Y. H. Geum Rathie, and Hj. Kim. A Note on Certain Laplace Transforms of Convolution-Type Integrals Involving Product of Two Generalized Hypergeometric Functions. Mathematical Problem in Engineering, 2021:8827275, 2021. [6] Hj. Kim. The intrinsic structure and properties of Laplace-typed integral transforms. Mathematical Problem in Engineering, 2017:1–8, 2017. REFERENCES 1216 [7] Hj. Kim. The solution of the heat equation without boundary conditions. Dynamic Systems and Applications, 27:653–662, 2018. [8] E. Kreyszig. Advanced Engineering Mathematics. Wiley, Singapore, 2013. [9] S. Supaknaree, K. Nonlaopon, and Hj. Kim. Further properties of Laplace-type integral transforms. 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