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- Leibniz integral rule. This important result is particularly useful in the differentiation of integral transforms. An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments of a random variable
- The Leibniz integral rule gives a formula for differentiation of a definite integral whose limits are functions of the differential variable, (1) It is sometimes known as differentiation under the integral sign

- General Leibniz rule. In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as Leibniz's rule). It states that if and are -times differentiable functions, then the product is also -times differentiable and its th derivative is given by
- In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, is of the form: where the partial derivative indicates that inside the integral, only the variation of f(•,u) with u is considered in taking the derivative. Simply
- The Leibniz Rule for an inﬁnite region I just want to give a short comment on applying the formula in the Leibniz rule when the region of integration is inﬁnite
- KC Border Differentiating an Integral: Leibniz' Rule 2. 2 The measure space case. This section is intended for use with expected utility, where instead if integrating with respect to a real parameter t as in Theorem 1, we integrate over an abstract probability space. So let Ω ,F,µ) be a measure space, let A ⊂ Rn be open
- Leibniz integral rule. then for x in ( x0, x1) the derivative of this integral is expressible as provided that f and its partial derivative fx are both continuous over a region in the form [ x0, x1] × [ y0, y1 ]. Thus under certain conditions, one may interchange the integral and partial differential operators
- Using R 1 0 e x2 = p ˇ 2, show that I= R 1 0 e x2 cos xdx= p ˇ 2 e 2=4 Di erentiate both sides with respect to : dI d = Z 1 0 e x2 ( xsin x) dx Integrate \by parts with u = sin x;dv = xe x2dx )du = cos xdx;v
- W are going to derive the Leibniz Rule for integrals in its whole form! It's one of the most powerful tools of integration, so be prepared! :)^ Quick note: The Integral I(x,t) is just in terms of.

- Leiniz rule). Using the de-nition (6), by the chain rule we obtain d( t) dt = @˚ @t + @˚ @a da dt + @˚ @b db dt: Using the results (1), (2), and (4), this gives d( t) dt = Z b(t) a(t) @f(z;t) @t dz+ f(b(t);t) db dt f(a(t);t) da dt which is the Leibniz rule. Drawing a picture (again remembering that an integral represents the area under a curve) will be helpful here
- Videos for Transport Phenomena course at Olin College This video describes the Leibniz Rule from calculus for taking the derivative of integrals where the limits of integration change with time
- ants, an expression for the deter
- If f and g are differentiable functions, then the chain rule explains how to differentiate the composite g o f.Now let us give separate names to the dependent and independent variables of both f and g so that we can express the chain rule in the Leibniz notation
- Leibniz's Rule: If f(x,y) is a well-behaved bi-variate function within the rectangle a<x<b, c<y<d, then we have: This useful formula, known as Leibniz's Rule, is essentially just an application of the fundamental theorem of calculus
- Under fairly loose conditions on the function being integrated, differentiation under the integral sign allows one to interchange the order of integration and differentiation. In its simplest form, called the Leibniz integral rule, differentiation under the integral sign makes the following equation valid under light assumptions on.

One of the useful consequences of the Leibniz rule for the fractional derivative of a product is a rule for evaluating the fractional derivative of a composite function. Let us take an analytic function φ ( t ) and f ( t ) = H ( t - a ), where H ( t ) is the Heaviside function The Leibniz rule is, together with the linearity, the key algebraic identity which unravels most of the structural properties of the differentiation. For this reason, in several situations people call derivations those operations over an appropriate set of functions which are linear and satisfy the Leibniz rule version of the Leibniz rule for differentiating an integral with variable limits of inte-gration, and using the generalized rule to find the Maxwellian and cavity fields in the source region. . I Leibniz's rule, as established in one dimension, introducesa correction term for each variable integration limit The Leibniz formula expresses the derivative on \(n\)th order of the product of two functions. Suppose that the functions \(u\left( x \right)\) and \(v\left( x \right)\) have the derivatives up to \(n\)th order. Consider the derivative of the product of these functions

Newton-Leibniz formula. From Encyclopedia of Mathematics \int\limits_a^bf(x)\,dx = F(b)-F(a). \end{equation} It is named after I. Newton and G. Leibniz, who both. Leibniz Rule for Di erentiating Products Formula to nd a high order derivative of product. Example: d9 dx9 (x sinx) = x d9 dx9 (sinx) + 9 d dx (x) d8 dx8 (sinx) + 9 Given the fact that he independently came up with the Fundamental Theorem of Calculus, it is no surprise that he also introduced the Leibniz Rule for Differentiating Under an Integral Sign, or simply the Leibniz Integral Rule. Alongside James Bernoulli, Leibniz showed us how to solve what we now call separable ordinary differential equations.

Solution. To calculate the derivative \({y^{\left( 5 \right)}}\) we apply the Leibniz rule. Let \(v = {x^3} + 2{x^2} + 3x\), \(u = {e^x}.\) Then the \(5\)th-order. Newton and Leibniz: the Calculus Controversy The History of Calculus The history of calculus does not begin with Newton and Leib-nizÕs Þndings. Their calculus was the culmination of centur ies of work by other mathematicians rather than an instant epiph any that came individually to them. Below is a summary of some o

Example of Leibniz rule for integration Can someone give me a concrete example of using Leibniz rule for evaluating the derivative with respect to x of f(x,y) where the limits of integration are functions of x On this page we'll be looking at the product **rule**, or the **Leibniz** **rule** as it's more eruditely called. As usual our goal is to visualize it. We'll also be looking at the complementary **rule** for integration: integration by parts

In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the form then for x in ( x 0, x 1) the derivative of this integral is thus expressibl ** Leibniz's rule (named after Gottfried Wilhelm Leibniz) may refer to one of the following: Product rule in differential calculus; General Leibniz rule**, a generalization of the product rule; Leibniz integral rule; The alternating series test, also called Leibniz's rule

** Stack Exchange network consists of 175 Q&A communities including Stack Overflow**, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers Leibnitz Rentals and Prices from 250 Top Sites. Compare and Book Now from $47 Free 2-Day Shipping w/Amazon Prim

- Leibnitz's rule. n (Mathematics) a rule for finding the derivative of the product of two functions
- If you look at regions 6 and 7 they are unbounded and the upper limit of the random variables goes to infinity. However, they have bounded limit in their expressions after the Leibniz rule is applied (1-beta)Cq
- 18.02, 18.02IS, 18.023 or even 18.024 at ESG Fall 2004 Leibniz's Formula Forthesenotes,thenotationwillbethatofSimmons,andallpageandequation referencesaretothatvolume. WhentheWebgetsbetter,alltypefaceswillbethesame. Untilthen,thefont in the ﬁgure uses a pointy-bottom vee that looks far too much like the Greek letter nu (ν)
- Abstract. We demonstrate that a violation of the Leibniz rule is a characteristic property of derivatives of non-integer orders. We prove that all fractional derivatives , which satisfy the Leibniz rule , should have the integer order , i.e. fractional derivatives of non-integer orders cannot satisfy the Leibniz rule

One of the most important one is the product rule, also known as Leibniz' rule, named after the 17th century German mathematician Gottfried Lebiniz, who discovered it. The product rule tells us how to compute the derivative of a product of two functions in terms of the original functions and their derivatives In calculus, the general Leibniz rule, [1] named after Gottfried Leibniz, generalizes the product rule (which is also known as Leibniz's rule). It states that if u and v are n-times differentiable functions, then product uv is also n-times differentiable and its nth derivative is given b which is two equations. We would then take two separate derivatives, and that's too much work. Instead, take derivatives of both sides of the equation, with respect to x: Since the derivative of a sum is the sum of the derivatives, and the derivative of a constant is 0. Leibniz - German philosopher and mathematician who thought of the universe as consisting of independent monads and who devised a system of the calculus independent of Newton (1646-1716) Gottfried Wilhelm Leibnitz, Gottfried Wilhelm Leibniz, Leibnitz

The Identity of Indiscernibles. The Identity of Indiscernibles is a principle of analytic ontology first explicitly formulated by Wilhelm Gottfried Leibniz in his Discourse on Metaphysics, Section 9 (Loemker 1969: 308). It states that no two distinct things exactly resemble each other The method of di erentiation under the integral sign, due to Leibniz in 1697 [4], concerns integrals The rule, called di erentiation under the integral sign, is. Prove Leibniz' rule using three properties. Ask Question 1 (x,t)dx $, the fundamental theorem of calculus, and the chain rule to prove Leibniz' rule:.

Leibniz' rule for differentiation of integrals 4/15/13 6.2 bjc (6.6) The order of the operations of integration and differentiation can be exchanged and so it is permissible to bring the derivative under the integral sign. We are interested in applications to compressible ﬂow and so from here on we will interpret the variable as time Teaching Discrete Mathematics and Computer Science (and now More!) via Primary Historical Sources. David is part of a team of mathematicians and computer scientists at this and other universities, who are applying this approach to the teaching of discrete mathematics (and now more!), broadly conceived Gottfried Leibniz: Causation. The views of Leibniz (1646-1716) on causation must stand as some of the more interesting in the history of philosophy, for he consistently denied that there is any genuine causal interaction between finite substances in a series of papers by T. J. Osler: Leibniz rule for fractional derivatives and an application to inﬁnite series, SIAM J. Appl. Math. 18, 658 (1970); The fractional derivative of a composite function, SIAM J. Math. Anal. 1

- tive notation for the derivative. A concept called di erential will provide meaning to symbols like dy and dx: One of the advantages of Leibniz notation is the recognition of the units of the derivative. For example, if the position function s(t) is expressed in me-ters and the time t in seconds then the units of the velocity function ds dt are.
- Engineering Mathematics Questions and Answers - Leibniz Rule - 1 Posted on July 13, 2017 by Manish This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on Leibniz Rule - 1
- A generalization of the Leibniz rule for derivatives R. DYBOWSKI School of Computing, University of East London, Docklands Campus, London E16 2RD e-mail: dybowski@uel.ac.uk I will shamelessly tell you what my bottom line is. It is placing balls into boxes . Gian-Carlo Rota (Indiscrete Thoughts) 1 Introductio

The German polymath Gottfried Wilhelm Leibniz occupies a grand place in the history of philosophy. He was, along with René Descartes and Baruch Spinoza, one of the three great 17th Century rationalists, and his work anticipated modern logic and analytic philosophy This rule is, indeed, due to Leibniz, although it was Johann Bernoulli who realized its broader implications, and there is an interesting story to its discovery. It is told in Chapter 3 of Families of Curves and the Origins of Partial Differentiation by Engelsman The Leibniz identity extends the product rule to higher-order derivatives. Algebra. Applied Mathematics. Calculus and Analysis. Discrete Mathematics. Foundations of.

- es the reality condition must satisfy the Yang-Baxter condition if the extension of the covariant derivative to tensor products is to satisfy the realit
- Leibniz rule of calculus is to be found in most advanced texts in mathematics, such as Wylie and Barrett (1982) or Abramowitz and Stegun (1965). The rule is represented by the equation: This rule is used in systems where integrations need to be performed over a time-dependent domain of integration
- Leibniz integral rule proof? Does anyone know how to get the Leibniz integral rule (a.k.a. differentiation under the integral sign)? I'm clueless. It..
- The two conditions of the test are met and so by the Alternating Series Test the series is convergent. It should be pointed out that the rewrite we did in previous example only works because \(n\) is an integer and because of the presence of the \(\pi\)
- This note provides a simple method to extend the usual Leibniz rule for higher derivatives of the product of two functions to several functions, which is within the reach of freshman calculus students
- Leibniz rule I discuss and solve a challenging integral. The method involves differentiation and then the solution of the resultant differential equation. The so-called Leibniz rule for differentiating integrals is applied during the process. Such an example is seen in 2nd-year university mathematics

(2016) Leibniz Rule and Fractional Derivatives of Power Functions. Journal of Computational and Nonlinear Dynamics 11 :3, 031014. (2016) Fractional-Order Euler-Lagrange Equation for Fractional-Order Variational Method: A Necessary Condition for Fractional-Order Fixed Boundary Optimization Problems in Signal Processing and Image Processing Why we study differential calculus. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked BARROW AND LEIBNIZ ON THE FUNDAMENTAL THEOREM OF THE CALCULUS Abstract. In 1693, Gottfried Whilhelm Leibniz published in the Acta Eruditorum a geometrical proof of the fundamental theorem of the calculus. During his notorious dispute with Isaac Newton on the development of the calculus, Leibniz denied any indebtedness to the work of Isaac Barrow

- The product rule for differentiation has analogues for one-sided derivatives. More explicitly, we can replace all occurrences of derivatives with left hand derivatives and the statements are true. Alternately, we can replace all occurrences of derivatives with right hand derivatives and the statements are true
- Further generalizations of the Leibniz rule are also given and are derived from a generalization of Taylor's series given previously by the author. It is shown that these new series are generalizations of Parseval's formula from the study of Fourier series
- It uses well known rules such as the linearity of the derivative, product rule, power rule, chain rule, so on. Additionally, `D` uses lesser known rules to calculate the derivative of a wide array of special functions. For higher order derivatives, certain rules like the general Leibniz product rule can speed up calculations
- If r = 2, the generalized Leibniz rule reduces to the plain Leibniz rule.This will be the starting point for the induction. To complete the induction, assume that the generalized Leibniz rule holds for a certain value of r; we shall now show that it holds for r + 1
- Leibniz here distinctly opposes identical truths as necessary, to truth connected with reason as contingent. Leibniz's New Essays Concerning the Human Understanding | John Dewey But, says Leibniz , there is something resisting, that to which Keppler gave the name inertia
- Leibniz was a strong believer in the importance of the product of mass times velocity squared which had been originally investigated by Huygens and which Leibniz called vis viva, the living force. He believed the vis viva to be the real measure of force, as opposed to Descartes's force of motion (equivalent to mass times velocity , or momentum )

- on the leibniz rule for random v ariables 12 where x 1 ,..., x n are distinct points in ( 0 , ∞ ) , must be conditionally negative deﬁnite, i.e. it is negative deﬁnite on the subspace X 0
- 18 Higher Derivative of the Product of Two Functions 18.1 Leibniz Rule about the Higher Order Differentiation Theorem 18.1.1 (Leibniz) When functions f()x and g()x are n times differentiable, the following expression holds
- d when he made his remarkable jump.] That rules out every sort of inﬂuence that one might think a created thing might have on something else. (I stress 'created' because of course I don't rule out God's affecting a monad.) Som
- In his reply Leibniz gave some details of the principles of his differential calculus including the rule for differentiating a function of a function. Newton was to claim, with justification, that not a single previously unsolved problem was solved.
- The Integration Theory of Gottfried Wilhelm Leibniz Zachary Brumbaugh History of Mathematics Rutgers, Spring 2000. The development of mathematics over the course of the last four millenia shows a steady though sometimes slow advance, with one mathematician's ideas greatly stimulating those of his successors
- We shall discuss generalizations of the Leibniz rule to more than one dimension. Such generalizations seem to be common knowledge among physicists, some dif- ferential geometers, and applied mathematicians who work in continuum mechanics

* Leibniz: Logic*. The revolutionary ideas of Gottfried Wilhelm Leibniz (1646-1716) on logic were developed by him between 1670 and 1690. The ideas can be divided into four areas: the Syllogism, the Universal Calculus, Propositional Logic, and Modal Logic Gottfried Leibniz From Wikipedia, the free encyclopedia. Gottfried Wilhelm von Leibniz ( July 1 , 1646 in Leipzig - November 14 , 1716 in Hannover ) was a German philosopher , scientist , mathematician , diplomat , librarian , and lawyer of Sorb descent Leibniz integral Rule Dr. Kumar Aniket University of Cambridge 1. Integrals 1.1. Leibniz integral Rule. •Diﬀerentiationundertheintegralsignwithconstantlimits You can enter expressions the same way you see them in your math textbook. Implicit multiplication (5x = 5*x) is supported. If you are entering the derivative from a mobile phone, you can also use ** instead of ^ for exponents. The interface is specifically optimized for mobile phones and small screens. Supported differentiation rules Gottfried Wilhelm Leibniz (also known as von Leibniz) was a prominent German mathematician, philosopher, physicist and statesman. Noted for his independent invention of the differential and integral calculus, Gottfried Leibniz remains one of the greatest and most influential metaphysicians, thinkers and logicians in history

Gottfried Wilhelm Leibniz: Gottfried Wilhelm Leibniz, German philosopher, mathematician, and political adviser, important both as a metaphysician and as a logician and distinguished also for his invention of the differential and integral calculus independent of Sir Isaac Newton The Holiness Problem. **Leibniz** argues that God is the author of all that is real and positive in the world, and that God is therefore also the author of all of privations in the world. It is a manifest illusion to hold that God is not the author of sin because there is no such thing as an author of a privation,.. Read Customer Reviews & Find Best Sellers. Free 2-Day Shipping w/Amazon Prime Leibniz Rule - DIFFERENTIATING UNDER THE INTEGRAL SIGN... An integral like R b a f ( x, t ) d x is a function of t , so we can ask about its t -derivative, assuming that f ( x, t ) is nicely behaved. The rule is: the t -derivative of the integral of f ( x, t ) is the integral of the t -derivative of f ( x, t ): (1.2) d d t Z b a f ( x,.. The Leibniz rule (algebra) and its meaning (Any Leibniz linear map of to itself is a Lie derivative along some vector field). Commutator of two vector fields (to be discussed more in the future). Push-forward of vector fields by smooth invertible maps

* Leibniz rule for both di erentiable and non-di erentiable functions*. The Leibniz rule holds for di erentiable functions with classical integer order derivative. Also the Leibniz rule still holds for non-di erentiable functions with a concise and essentially local de nition of fractional derivative We present below continually more complex extensions of the Leibniz rule. Case I, The simplest integral analog of the Leibniz rule is. (1.1) d:u{z)v(z) = f {^jDrau(z)d:v(z)dw, where () = T(a + l)/r(a: — co + l)T(co + 1), and a is any real or complex number. Notice that we integrate over the order of the derivatives

Show transcribed image text Leibniz' Rule 5x 58. + t dt 2+3x 1-2x 60 61. dt 62 dt cos(2t) dt) 67. xoCos(3) do) dt 0 Expert Answer This problem has been solved Continuing Leibniz rule... Remember we wanted to solve problems like d dx Z. v(x) u(x) f(x;t)dt So far have only considered the case u(x) = a and v(x) = b were just constants Let us now look at the problem when f(x;t) is only a function of t, f(t) dI dx = d dx Z. v(x) u(x In calculus , Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz , states that for an integral of the form where − ∞ a ( x ) , b ( x ) ∞ {\\displaystyle -\\infty , the derivative of this integral is expressible as where the partial derivative indicates that inside the integral, only the variation of f(x, t) with t is considered in taking the derivative

The Chain Rule in Leibniz Notation. the two statements of the chain rule do mean the same thing. We can remember the chain rule in Leibniz notation because it looks like a nice fraction equation where the dy terms cancel: This may or may not be what's actually going on, but it works for our purposes and it's a great memory aid. There are.. Dear pradyot. Leibniz Integral Rule. The Leibniz integral rule gives a formula for differentiation of a definite integral whose limits are functions of the differential variable, where the partial derivative of f indicates that inside the integral only the variation of ƒ ( x, α ) with α is considered in taking the derivative Proof. If you know Newton's binomial formula, you will notice that these 2 formulas (Newton's and Leibniz') are very similar, because they work in the same way : induction is the same Then, since chain rule maps a function to a sum of products of functions, what makes the job for subsequent expansions is the Leibniz' s law (derivation of the product of an algebra over a field)

Lecture 5 Max-Min Values First Derivative Test Second derivative test Global Extreme Lagrange multipliers Two constraints Leibniz' rule Leibniz' integration rule Example Example Find d dx cos x sin x cosh t 2 dt. Solution Applying Leibniz's integral rule one has d dx cos x sin x cosh t 2 dt cos x sin x x cosh t 2 dt sin x cosh cos 2 x. Best Answer: Note: I only have worked with Leibniz Rule for single integrals. Below is a calculation only using that with a little of Fubini's Theorem Leibniz formula, por exemplo, em varios textos logicos, seu famoso principio de substitutibilidade salva veritate, o qual afirma que dois termos sao identicos em significado quando podem ser substituidos um pelo outro nas sentencas sem que isso acarrete nenhuma alteracao do valor de verdade destas Leibniz Notation for Derivatives Page 3 of 3 Using Leibniz Notation to Keep Track of Pieces of a Long Derivative Leibniz notation is also exceptionally good as keeping track of what is happening during the derivative of a complicated function (one that involves a combination of product rule, quotient rule, chain rule, etc). Example Di erentiate. Leibniz's Formula - Differential equation How to do this difficult integral? Help with differentiation maths in medicine uni peaakk Total confusion with chain rule The Leibnitz Formula Edexcel A level Leibnitz Theorem HELP!!!

- imizes the value of the following integral I(x) = xZ+1 x ln (u)du: Solution: Using the leibniz.
- $\begingroup$ As for why the two derivatives are made to agree on scalars, my response is that this is a desired property of the covariant derivative that we enforce, just like linearity and obeying the Leibniz (product) rule. (Perhaps someone has a more enlightening reason.
- A Leibniz integral rule for three dimensions is: [1] where: F ( r, t ) is a vector field at the spatial position r at time t Σ is a moving surface in three-space bounded by the closed curve ∂Σ d A is a vector element of the surface Σ d s is a vector element of the curve ∂Σ v is the velocity of movement of the region
- have obtained a new generalized Leibniz rule and a corresponding integral analogue for the fractional derivatives of the product of two functions. In this paper, we apply the new transformation formula on the classical generalized Leibniz rule and the corresponding integral analogue due to Osler and on those established by the authors
- Home > Mathematics > Calculus > Engineering Mathematics > Differentiate Under Integral Signs: Leibniz Rule Lecture Details: This presentation shows how to differentiate under integral signs via. Leibniz rule. Many examples are discussed to illustrate the ideas. A proof is also given of the most basic case of Leibniz rule
- Similar to the Leibniz rule for the gradient, there is a Leibniz rule for the divergence of a product of a scalar ﬁeld S(x,y,z) and a vector ﬁeld V(x,y,z)

- The product rule of differential calculus is still called Leibniz's law. In addition, the theorem that tells how and when to differentiate under the integral sign is called the Leibniz integral rule 13. All mathematicians weren't the most intelligent people on earth. For them , THEIR I CAN WAS MORE IMPORTANT THAN THEIR IQ 14
- The product rule is a formal rule for differentiating problems where one function is multiplied by another. The rule follows from the limit definition of derivative and is given by . Remember the rule in the following way. Each time, differentiate a different function in the product and add the two terms together
- A fórmula de Leibniz, em referência a Gottfried Wilhelm Leibniz, é uma fórmula que expressa a derivada de uma integral como a integral de uma derivada. Explicitamente, seja uma função de x dada pela integral definida: ∫ (,
- Definition of Leibnitz's rule from the Collins English Dictionary The dash ( - ) A spaced dash(i.e. with a single space before and after it) is used: at the beginning and end of a comment that interrupts the flow of a sentence
- Contrary to Pascal, Leibniz (1646-1716) successfully introduced a calculator onto the market. It is designed in 1673 but it takes until 1694 to complete. The calculator can add, subtract, multiply, and divide
- g the theorem that students of partial differential equations know as the Leibniz rule. Theorem 2
- Payne lost his driving licence a year ago for drink-driving. Payne lost his driving license a year ago for drink-driving. As she passed the library door, the phone began to ring. As she past the library door, the phone began to ring. Strong winds blew away most of the dust. Strong winds blue away.

In this paper, we prove that unviolated simple Leibniz rule and equation for fractional-order derivative of power function cannot hold together for derivatives of orders α ≠ 1. To prove this statement, we use an algebraic approach, where special form of fractional-order derivatives is not applied Consistent with the liberal views of the Enlightenment, Leibniz was an optimist with respect to human reasoning and scientific progress. Although he was a great reader and admirer of Spinoza, Leibniz, being a confirmed deist, rejected emphatically Spinoza's pantheism Looking for Leibniz, Gottfried Wilhelm Von? Find out information about Leibniz, Gottfried Wilhelm Von. Born July 1, 1646, in Leipzig; died Nov. 14, 1716, in Hanover

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