Malmsten's Proof of the Integral Theorem

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property of being two telescoping sums which in the limit becomes the clockwiseintegration above. The counter clockwise integral is reached via a similarapproach.In more detail, the sum can be written asS n =r=n ∑r=1[(x r − x r−1 )p(x r−1 , y r−1 ) + (y r − y r−1 )q(x r−1 , y r−1 )] ,and each term can be rewritten as∫ xrx 0(x r − x r−1 )p(x r−1 , y r−1 ) + (y r − y r−1 )q(x r−1 , y r−1 ) =∫ xr−1∫ yrp(x, y r )dx− p(x, y r−1 )dx+ q(x 0 , y)dy+(x r −x r−1. )λ r +(y r −y r−1 )σ r ,x 0y r−1where λ r and σ r are error terms approaching 0 in the limit. To prove thisequality Malmsten makes several calculations with Riemann-sums, makesuse of the mean value-theorem for integrals and uses expression (7) in differentialform. He does use the fact that the functions are continuous, althoughhe doesn’t mention (or realize) it. For instance he writes“Men nu är tillika(y r − y r−1 )[q(x 0 , y r−1 ) + γ r ] =∫ yry r−1q(x 0 , y)dy + (y r − y r−1 )σ ′ r(der σ ′ r convergerar mot 0 på samma gång som differenserna(17)), hvilket insatt...”This equality can be achieved by applying the mean value-theorem for integralson the right hand side integral and using continuity for the functionq when substituting a value c (from the mean-value theorem for integrals)between y r−1 and y r with y r−1 . Consequently this gives rise to an error termthat is added to σ ′ r producing a new error term γ r which converges to 0 inthe limit. This is only one of several passages where ’common knowledge’theorems together with the conditions of the proof is used.Although Malmsten had come up with the condition (7), he didn’t useit in Greens formula. Probably he didn’t know about it, even though it waspublished by W. Thompson (Lord Kelvin) in Crelle’s Journal in 1850 [2].In fact, George Green (1793-1841) had published it, at his own expense, asearly as 1828 [2]. But the fact that Green was a physicist together with alacking communication may have been some of the reasons why the theoremremained unknown for more than 20 years.16
In the next part Malmsten uses the Theorem II twice in a proof of asimilar theorem for complex valued functions. I.e. he shows that the sumS n = (z 1 − z 0 )f(z 0 ) + (z 2 − z 1 )f(z 1 ) + . . . + (Z − z n−1 )f(z n−1 ),where z = x + iy and(f synectic) as n → ∞ and the differencesf(z) = ϕ(x, y) + iψ(x, y), (9)z 1 − z 0 , z 2 − z 1 , . . . , Z − z n−1i.e. the differencesx 1 − x 0 , x 2 − x 1 , . . . , X − x n−1 ,y 1 − y 0 , y 2 − y 1 , . . . , Y − y n−1decrease indefinitely, is independent of n as well as of the valuesz 1 , z 2 , z 3 , . . . , z n−1 .To prove this Malmsten constructs the sumS n = P n + iQ nwhereP n = (x 1 − x 0 ) · ϕ(x 0 , y 0 ) − (y 1 − y 0 ) · ψ(x 0 , y 0 )+(x 2 − x 1 ) · ϕ(x 1 , y 1 ) − (y 2 − y 1 ) · ψ(x 1 , y 1 )+ . . .+(X − x n−1 ) · ϕ(x n−1 , y n−1 ) − (Y − y n−1 ) · ψ(x n−1 , y n−1 )andQ n = (x 1 − x 0 ) · ψ(x 0 , y 0 ) + (y 1 − y 0 ) · ϕ(x 0 , y 0 )+(x 2 − x 1 ) · ψ(x 1 , y 1 ) + (y 2 − y 1 ) · ϕ(x 1 , y 1 )+ . . .+(X − x n−1 ) · ψ(x n−1 , y n−1 ) + (Y − y n−1 ) · ϕ(x n−1 , y n−1 )17
Page 1 and 2: Malmsten's Proof of the Integral Th
Page 3 and 4: AbstractIn this report we consider
Page 5 and 6: 1 IntroductionDuring the eighteenth
Page 7 and 8: 3 A short History of Complex Analys
Page 9 and 10: ecognition after 1831, when Carl Fr
Page 11 and 12: seems to have taken an interest in
Page 13 and 14: After that, Cauchy indicates the in
Page 15: 5.1 The Insufficiencies in Cauchy
Page 20 and 21: and applies Theorem II.It is notabl
Page 22 and 23: that Cauchy didn’t mention this p
Page 24: [7] Smithes, F. (1997). Cauchy and

Malmsten's Proof of the Integral Theorem

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