The Birth of Calculus: 1666 & 1675
Jeremy Gray (1986. Aired 1992年07月27日).
Calculus: One of the Most Successful Technologies (1:42:47)
by Keith Devlin (Stanford, 2012年12月11日).
Let's give the traditional approach first. This will be complemented by an abstract glimpse of the bigger picture, which is more closely related to the way people actually use derivatives, once they are familiar with them.
For a given real-valued function f of a real variable, consider the slope (m) of its graph at some point. That is to say, some straight line of equation y = mx+b (for some irrelevant constant b) is tangent to the graph of f at that point. In some definite sense, mx+b is the best linear approximation to f(x) when x is close to the point under consideration...
The tangent line at point x may be defined as the limit of a secant line intersecting a curve at point x and point x+h, when h tends to 0. When the curve is the graph of f, the slope of such a secant is equal to [ f(x+h)-f(x) ] / h, and the derivative (m) at point x is therefore the limit of that quantity, as h tends to 0.
The above limit may or may not exist, so the derivative of f at point x may or may not be defined. We'll skip that discussion. The popular trivia question concerning the choice of the letter "m" to denote the slope of a straight line (in most US textbooks) is discussed elsewhere.
Way beyond this introductory scope, we would remark that the quantity we called h is of a vectorial nature (think of a function of several variables), so the derivative at point x is in fact a tensor whose components are called partial derivatives.
Also beyond the scope of this article are functions of a complex variable, in which case the above quantity h is simply a complex number, and the above division by h remains thus purely numerical (albeit complex). However, a complex number h (a point on the plane) may approach zero in a variety of ways that are unknown in the realm of real numbers (points on the line). This happens to severely restrict the class of functions for which the above limit exists. Actually, the only functions of a complex variable which have a derivative are the so-called analytic functions [essentially: the convergent sums of power series].
The above is the usual way the concept of derivative is often introduced. This traditional presentation may be quite a hurdle to overcome, when given to someone who may not yet be thoroughly familar with functions and/or limits. Having defined the derivative of f at point x, we define the derivative function g = f ' = D( f ) of the function f, as the function g whose value g(x) at point x is the derivative of f at point x.
We could then prove, one by one, the algebraic rules listed in the first lines of the following table. These simple rules allow most derivatives to be easily computed from the derivatives of just a few elementary functions, like those tabulated below (the above theoretical definition is thus rarely used in practice):
One abstract approach to the derivative concept would be to bypass (at first) the relevance to slopes, and study the properties of some derivative operator D, in a linear space of abstract functions endowed with an internal product (´), where D is only known to satisfy the following two axioms (which we may call linearity and Leibniz' law, as in the above table):
For example, the product rule imposes that D(1) is zero [in the argument of D, we do not distinguish between a function and its value at point x, so that "1" denotes the function whose value is the number 1 at any point x]. The linearity then imposes that D(a) is zero, for any constant a. Repeated applications of the product rule give the derivative of x raised to the power of any integer, so we obtain (by linearity) the correct derivative for any polynomial. (The two rules may also be used to prove the chain rule for polynomials.)
A function that has a derivative at point x (defined as a limit) also has arbitrarily close polynomial approximations about x. We could use this fact to show that both definitions of the D operator coincide, whenever both are valid (if we only assume D to be continuous, in a sense which we won't make more precise here).
Convolution products in the theory of distributions.The above abstarction is directly accessible at the elementary level. Beyond that, a very fruitful viewpoint is totally different, In the Theory of Distributions, as a pointwise product like the above (´) is not even defined, whereas everything revolves around the so-called convolution product (*), which has the following strange property concerning the operator D:
D( u * v ) = D(u) * v = u * D(v)
To differentiate a convolution product (u*v), differentiate either factor!
When the logarithm of a function is well-defined, the logarithmic derivative is the derivative of the logarithm.
However, the logarihmic derivative of a nonzero differentiable function is always well-defined, even when the logarithm isn't (as is the case for negative functions or for complex-valued functions, where the logarithm could only be defined as a multivalued function).
Wikipedia : Logarithmic derivative
Once known as Barrow's rule, it states that, if f is the derivative of F, then:
In this, if f and F are real-valued functions of a real variable, the right-hand side represents the area between the curve y = f (x) and the x-axis (y = 0), counting positively what's above the axis and negatively [negative area!] what's below it.
Any function F whose derivative is equal to f is called a primitive of f (all such primitives simply differ by an arbitrary additive constant, often called constant of integration). A primitive function is often called an indefinite integral (as opposed to a definite integral which is a mere number, not a function, possibly obtained as the difference of the values of the primitive at two different points). The usual indefinite notation is:
F(x) = ò f (x) dx
At a more abstract level, we may also call Fundamental Theorem of Calculus the generalization of the above expressed in the language of differential forms, which is also known as Stokes' Theorem, namely:
Fundamental Theorem of Calculus (Theorem of the Day #2) by Robin Whitty
That function is the real part of a complex function of a real variable:
(cos 2x + i sin 2x) e 3x = e i (2x) e 3x = e (3+2i) x
Since the derivative of exp(a x) / a is exp(a x) we obtain, conversely:
ò e(3+2i) x dx = e(3+2i) x / (3+2i) = e3x (cos 2x + i sin 2x) (3-2i) / 13
The relation we were after is obtained as the real part of the above:
ò cos(2x) e 3x dx = (3 cos 2x + 2 sin 2x) e 3x / 13
This method was first published in 1715 by Brook Taylor (1685-1731).
The product rule states that the derivative (uv)' of a product of two functions is u'v+uv'. When the integral of some function f is sought, integration by parts is a minor art form which attempts to use this backwards, by writing f as a product u'v of two functions, one of which (u') has a known integral (u). In which case:
ò f dx = ò u'v dx = uv - ò uv' dx
This reduces the computation of the integral of f to that of uv'. The tricky part, of course, is to guess what choice of u would make the latter simpler...
The choice u' = 1 (i.e., u = x and v = f ) is occasionally useful. Example:
ò ln(x) dx = x ln(x) - ò (x/x) dx = x ln(x) - x
Another classical example pertains to Laplace transforms ( p > 0 ) and/or Heaviside's operational calculus, where all integrals are understood to be definite integrals from 0 to +¥ (with a subexponential function f ):
ò f '(t) exp(-pt) dt = - f (0) + p ò f (t) exp(-pt) dt
Last but not least, integration by part may produce similar integral remainders. Iterating such a process may generate famous summations, like Taylor's expansion, Darboux's formula, Euler-Maclaurin formula...
Integration by parts
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Operational calculus
Integration by Parts
by Robin Whitty
(Theorem of the Day #219).
The Wallis integral of order n is the definite integral:
As the integrand is a pointwise decreasing function of the integer n, these integrals form a decreasing sequence. Trivially, I0 = p/2 and I1 = 1. Let's integrate by parts the Wallis integral of order n+2 when n ≥ 0 :
On the right-hand-side, the square bracket vanishes and we may replace cos2 t by 1 - sin2 t to obtain In+2 = (n+1) ( In - In+2 ). Therefore:
(n+2) In+2 = (n+1) In
So, we have a recurrence for even orders and another one for odd orders.
Therefore, I2n I2n+1 = p / (4n+2). As the sequence is decreasing, the sandwich theorem implies that consecutive terms are asymptotically equivalent. This makes I2n2 equivalent to p/4n. In other words:
In ~ ( p/2n )1/2
Combined with the above expression for I2n (or I2n+1) that equivalence is what allowed James Stirling (1692-1770) to derive his famous formula for the asymptotic equivalent of n! (1730).
One popular way to present the fact that the ratio I2n+1 / I2n tends to one as n tends to infinity is to state that the following product is convergent :
Wallis' integrals
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Wallis product (1655)
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John Wallis (1616-1703)
A quantum-mechanical derivation of the Wallis formula for p (2015).
Wallis product formula (5:42)
by Jens Fehlau (Flammable Maths, 2019年05月28日).
The Wallis product for pi, proved geometrically (25:26)
by Grant Sanderson (3blue1brown, 2018年04月20日).
Amazing formula for pi: the Wallis product (11:56)
by Presh Talwalkar
(MindYourDecisions, 2016年10月12日).
Choose the coordinate axes so that your parabola has equation y = x2/2p for some constant parameter p. The length element ds along the parabola is such that (ds)2 = (dx)2 + (dy)2, or ds/dx = Ö(1+(dy/dx)2) = Ö(1 + x2/p2). The length s of the arc of parabola from the apex (0,0) to the point (x, y = x2/2p) is simply the following integral of this (in which we may eliminate x or p, using 2py = x2 ).
For a symmetrical arc extending on both sides of the parabola's axis, the length is 2s (twice the above). If needed, the whole "perimeter" is 2s+2x.
Let's express all distances in feet (a mile is 5280 ft). Using the notations of the previous article, 2x = 5280, 2s = 5281, u = x/p = 2y/x = y/1320
For small values of u, the right-hand side is roughly 1+u2/6. Solving for u the equation thus simplified, we obtain Ö(6/5280) = 0.033709993123... The height y is thus roughly equal to that quantity multiplied by 1320 ft , which is nearly 44.4972 ft.
This approximation is valid for any type of smooth enough curve. It can be refined for the parabolic case using successive approximations to solve for u the above equation. This yields u = 0.0337128658566... which exceeds the above by about 85.2 ppm (ppm = parts per million) for a final result of about 44.5010 ft. Our first approximation would've satisfied any engineer before the computer era. The solution is best expressed as 44½ ft.
Answer : Surprisingly, just about 28.00857 m...
Under its own weight, a uniform cable without any rigidity (a "chain") would actually assume the shape of a catenary. In a coordinate system with a vertical y-axis and centered on its apex, the catenary has the following cartesian equation:
y/a = ch (x/a) -1 = ½ (ex/a - 2 + e-x/a ) = 2 sh2 (x/2a)
Measured from the apex at x = y = 0, the arclength s along the cable is:
s = a sh (x/a)
Those formulas are not easy to work with, unless the parameter a is given. For example, in the case at hand (a 28 m span with a 0.3 m sag) we have:
x = 14 y = 0.3
So, we must solve for a (numerically) the transcendantal equation:
0.3 / a = 2 sh2 (7/a)
This yields a = 326.716654425... Therefore :
2s = 2a sh (14 / a) = 28.00856959...
Thus, an 8.57 mm slack produces a 30 cm sag for a 28 m span.
The parameter a is the radius of curvature at the curve's apex. If it's large enough, we may find a good approximation to the relevant transcendental equation by equating the sh function to its (small) argument:
y/a =
2 sh2 (x/2a)
yields
a » x2 / 2y
whereby s = a sh (x/a)
»
x ( 1 + x2 / 6a2 )
»
x ( 1 + 2y2 / 3x2 )
This gives 2s » 2x ( 1 + 8/3 (y/2x)2 ) = 28.0085714... in the above case. This is indeed a good approximation to the aforementioned exact result.
If we plug the values x = 14 and y = 0.3 in the above formula for the exact length of a parabolic arc, we obtain: 2s = 28.0085690686...
A thin circular arc of width 2x and of height y has a length
In fact, all smooth approximations to a flat enough catenary will have a comparable precision, because this is what results from equating a curve to its osculating circle at the lowest point. The approximative expression we derived above in the case of the catenary is indeed quite general:
As a wheel of unit radius rolls on the x-axis, the trajectory of a point on its circumference is a cycloid, whose parametric equation is :
x = t - sin(t) y = 1 - cos(t)
This curve was first studied by Marin Mersenne (1588-1648) around 1615. The parameter t is the abscissa of the wheel's center. In the first revolution of the wheel (one arch of the cycloid), t goes from 0 to 2p.
The length of one full arch of a cycloid ("cycloidal arch") was first worked out in the 17th century by Evangelista Torricelli (1608-1647), just before the advent of the calculus. Let's do it now with modern tools:
Calling s the curvilinear abscissa (the length along the curve), we have:
(ds)2 =
(dx)2 + (dy)2 =
[(1-cos(t))2 + (sin(t))2](dt)2
therefore:
(ds/dt)2 = 2 - 2 cos(t) = 4 sin2(t/2)
so, if 0 ≤ t ≤ 2p:
ds/dt = 2 sin(t/2) ≥ 0
The length of the whole arch is the integral of this when t goes from 0 to 2p and it is therefore equal to 8, [since the indefinite integral is -4 cos(t/2)]. On the other hand, the length of the trajectory of the wheel's center (a straight line) is clearly 2p (the circumference of the wheel). In other words, the trajectory of a point on the circumference is 4/p times as long as the trajectory of the center, for any whole number of revolutions (that's about 27.324% longer, if you prefer).
The ratio you asked for is the reciprocal of that, namely p/4 (which is about 0.7853981633974...), the ratio of the circumference of the wheel to the length of the cycloidal arch. However, the result is best memorized as:
"The length of a cycloidal arch is 4 times the diameter of the wheel."
An obvious change of variable is to introduce y = tan x [ dy = (1+y2 ) dx ], so the integrand becomes y1/3 dy / (1+y2 ). This suggests a better change of variable, namely: z = y2/3 = (tan x)2/3 [ dz = (2/3)y-1/3 dy ], which yields z dz = (2/3)y1/3 dy, and makes the integrand equal to the following rational function of z, which may be integrated using standard methods (featuring a decomposition into 3 easy-to-integrate terms):
(3/2) z dz / (1+z3 ) =As (1-z+z2 ) is equal to the positive quantity ¼ [(2z - 1)2 + 3] , we obtain:
ò(tan x)1/3 dx
=
¼ ln(1-z+z2 )
+
¼Ö3 arctg((2z-1)/Ö3)
- ½ ln(1+z)
where z stands for | tan x | 2/3
We assume all distances are in meters.
When the particle is at a negative abscissa x, the (negative) slope of the line in question is y/x = Ö(-x)/x and the corresponding (negative) angle is thus:
a = arctg(Ö(-x)/x)[In this, "arctg" is the "Arctangent" function, which is also spelled "atan" in US textbooks.] Therefore, a varies with x at a (negative) rate:
da/dx = -1/(2´Ö(-x)(1-x)) (rad/m)If x varies with time as stated, we have dx/dt = -8 m/s, so the angle a varies with time at a (positive) rate:
da/dt = 4/(Ö(-x)(1-x)) (rad/s)
When x is -4 m, the rate dA/dt is therefore 4/(Ö4 ´5) rad/s = 0.4 rad/s.
The angle a, which is always negative, is thus increasing at a rate of 0.4 rad/s when the particle is 4 meters to the left of the origin (rad/s = radian per second).
The curves intersect when f(x) = g(x), which translates into x3 - 10x - 3 = 0. This cubic equation factors nicely into (x + 3) (x2 - 3x - 1) = 0 , so we're faced with only a quadratic equation...
To find if there's a "trivial" integer which is a root of a polynomial with integer coefficients [whose leading coefficient is ±1], observe that such a root would have to divide the constant term. In the above case, we only had 4 possibilities to try, namely -3, -1, +1, +3.
The abscissas A < B < C of the three intersections are therefore:
A = -3 , B = ½ (3 - Ö13) and C = ½ (3 + Ö13)
The best thing to do for a "figure 8", like the one at hand, is to compute the (positive) areas of each of the two lobes. The understanding is that you may add or subtract these, according to your chosen orientation of the boundary:
The area we're after is thus either the sum (±51.356...) or the difference (±18.952...) of these two, depending on an ambiguous boundary orientation...
If you don't switch curves at point B, the algebraic area may also be obtained as the integral of g(x)-f(x) from A to C (up to a change of sign).
The area surrounded by a closed planar curve may be defined in general terms, even when the curve does cross itself many times...
The usual algebraic definition of areas depends on the orientation (clockwise or counterclockwise) given to the closed boundary of a simple planar surface. The area is positive if the boundary runs counterclockwise around the surface, and negative otherwise (by convention, the positive direction of planar angles is always counterclockwise). In the case of a simple closed curve [without any multiple points] this is often overlooked, since we normally consider only whichever orientation of the curve makes the area of its interior positive...
The clear fact that there is such an "interior" bounded by any given closed planar curve is known as "Jordan's Theorem". It's a classical example of an "obvious" fact with a rather intricate proof.[画像: A curve with many double points.]
However, when the boundary has multiple points (like the center of a "figure 8"), there may be more than two oriented boundaries for it, since we may have a choice at a double point: Either the boundary crosses itself or it does not (in the latter case, we make a sharp turn, unless there's an unusual configuration about the intersection). Not all sets of such choices lead to a complete tracing of the whole loop.
It's nice to consider a given oriented closed boundary as a projection of a three-dimensional loop whose apparent area is defined as a path integral.
Winding number
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Nonzero-rule
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Even-odd rule
Numericana :
Surface rea bordered by a oriented loop drawn on the surface of an ellipsoid
The apparent area bordered by an oriented 3D loop is
a direction dotted into vectorial area.
As shown below, a high-order linear DE can be reduced to a system of first-order linear differential equations in several variables. Such a system is of the form:
X' = dX/dt = AX + B
X is a column vector of n unknown functions of t. The square matrix A may depend explicitely on t. B is a vector of n explicit functions of t, called forcing terms. The associated homogeneous system is obtained by letting B = 0. For a nonconstant A, it may be quite difficult to find n independent solutions of this homogeneous system (an art form in itself) but, once you have them, a solution of the forced system may be obtained by generalizing to n variables the method (called "variation of parameters") commonly used for a single variable. Let's do this using only n-dimensional notations:
The fundamental object is the square matrix W formed with the n columns corresponding to the n independent solutions of the homogeneous system. Clearly, W itself verifies the homogeneous equation:
W' = AW
It's an interesting exercise in the manipulation of determinants to prove that det(W)' = tr(A) det(W) (HINT: Differentiating just the i-th line of W gives a matrix whose determinant is the product of det(W) by the i-th component in the diagonal of the matrix A). Since det(W), the so-called "Wronskian", is thus solution of a first-order linear DE, it's proportional to the exponential of some function and is therefore either nonzero everywhere or zero everywhere. (Also, the Wronskians for different sets of homogeneous solutions must be proportional.)
Homogeneous solutions that are linearly independent at some point are therefore independent everywhere and W(t) has an inverse for any t. We may thus look for the solution X to the nonhomogeneous system in the form X = WY :
AX + B = X' = W'Y + WY' =
AWY + WY'
= AX + WY'
Therefore, B = WY'
So, Y is simply obtained by integrating W-1 B and the general solution of the forced system may be expressed as follows, with a constant vector K (whose n components are the n "constants of integration"). This looks very much like the corresponding formula for a single variable :
A linear differential equation of order n has the following form (where ak and b are explicit functions of t):
x(n) + an-1 x(n-1) + ... + a3 x(3) + a2 x" + a1 x' + a0 x = b
This reduces to the above system X' = AX + B with the following notations :
The first n-1 components in the equation X' = AX+B merely define each component of X as the derivative of the previous one, whereas the last component expresses the original high-order differential equation. Now, the general discussion above applies fully with a W matrix whose first line consists of n independent solutions of the homogeneous equation (each subsequent line is simply the derivative of its predecessor).
We need not work out every component of W-1 since we're only interested in the first component of X... The above boxed formula tells us that we only need the first component of W(t)W-1(u)B(u) which may be written G(t,u)b(u), by calling G(t,u) the first component of W(t)W-1(u)Z, where Z is a vector whose component are all zero, except the last one which is one.
G(t,u) is called the Green function associated to the given homogeneous equation. It has a simple expression (given below) in terms of a ratio of determinants computed for independent solutions of the homogeneous equation. (Such an expression makes it easy to prove that the Green function is indeed associated to the equation itself and not to a particular set of independent solutions, as it is clearly invariant if you replace any solution by some linear combination in which it appears with a nonzero coefficient.)
For a third-order equation with homogeneous solutions A(t), B(t) and C(t), the expression of the Green function (which generalizes to any order) is:
It's also a good idea to define G(t,u) to be zero when u>t, since such values of G(t,u) are not used in the integral ò t G(t,u) b(u) du. This convention allows us to drop the upper limit of the integral, so we may write a special solution of the inhomogeneous equation as the definite integral (from -¥ to +¥, whenever it converges): ò G(t,u) b(u) du.
If this integral does not converge (the issue may only arise when u goes to -¥), we may still use this formal expression by considering that the forcing term b(u) is zero at any time t earlier than whatever happens to be the earliest time we wish to consider. (This is one unsatisfying way to reestablish some kind of fixed arbitrary lower bound for the integral of interest when the only natural one, namely -¥, is not acceptable.)
In the case of the equation x''' - 3x" + 4x = exp(2t), three independent solutions are A(t) = exp(-t), B(t) = exp(2t), and C(t) = t exp(2t). This makes the denominator in the above (the "Wronskian") equal to 9 exp(3u) whereas the numerator is (3t-3u-1)exp(u+2t)+exp(4u-t). With those values, the integral of G(t,u)exp(2u)(u)du when u goes from 0 to t turns out to be equal to f(t) = [ (9t2-6t+2)exp(2t) - 2 exp(-t) ]/54, which is therefore a special solution of your equation. The general solution may be expressed as:
x(t) = (a + bt + t2/6) exp(2t) + c exp(-t) [ a, b and c are constant ]
Clearly, this result could have been obtained without this heavy artillery: Once you've solved the homogeneous equation and realized that the forcing term is a solution of it, it is very natural to look for an inhomogeneous solution of the form z exp(2t) and find that z"=1/3 works. That's far less tedious than computing and using the associated Green's function. However, efficiency in this special case is not what the question was all about...
The above may be dealt with using the elegant idea of convolution products among distributions. The notorious Theory of Distributions occurred to the late Laurent Schwartz (1915-2002) "one night in 1944". [画像: Fields Medal ]
For this, he received the first Fields Medal ever awarded to a Frenchman, in 1950. Schwartz taught at Polytechnique from 1958 to 1980. (He taught me functional analysis in the Fall of 1977.)
A linear differential equation with constant coefficients (an important special case) may be expressed as a convolution a * x = b. The convolution operator * is bilinear, associative and commutative. Its identity element is the Delta distribution d (dubbed Dirac's "function").
Loosely speaking, the Delta distribution d would correspond to a "function" whose integral is 1, but whose value at every point except zero is zero. The integral of an ordinary function which is zero almost everywhere would necessarily be zero. Therefore, the d distribution cannot possibly be an ordinary function: Convolutions must be put in the proper context of the Theory of Distributions. A strong case can be made that the convolution product is the notion that gives rise to the very concept of distribution.
Distributions had been used loosely by physicists for a long time, when Schwartz finally found a very simple mathematical definition for them: Considering a (very restricted) space D of so-called test functions, a distribution is simply a linear function which associates a scalar to every test function.
Although other possibilities have been studied (which give rise to less general distributions) D is normally the so-called Schwartz space of infinitely derivable functions of compact support These are perfectly smooth functions vanishing outside of a bounded domain, like the function of x which is exp(-1 / (1-x 2 )) in [-1,+1] and 0 elsewhere.
What could be denoted f(g) is written f*g instead. This hint of an ultimate symmetry between the rôles of f and g is fulfilled by the following relation, which holds whenever the integral exists for ordinary functions f and g.
[f*g](t) = ò f(t-u)g(u) du
This relation may be used to establish commutativity (switch the variable to v = t-u, going from +¥ to -¥ when u goes from -¥ to +¥). The associativity of the convolution product is obtained by figuring out a double integral.
Convolutions have many stunning properties. In particular, the Fourier transform of the convolution product of two functions is the ordinary product of their Fourier transforms. Another key property is that the derivative of a convolution product may be obtained by differentiating either one of its factors:
(f*g)' = f'*g = f*g'
This means the derivatives of a function f can be expressed as convolutions, using the derivatives of the d distribution (strange but useful beasts):
f = d * f f' = d' * f f'' = d'' * f etc.
If the n-th order linear differential equation discussed above has constant coefficients, we may write it as f*x = b by introducing the distribution
f = d(n) + an-1 d(n-1) + ... + a3 d(3) + a2 d" + a1 d' + a0 d
Clearly, if we we have a function such that f*g=d, we will obtain a special solution of the inhomogeneous equation as g*b. If you translate the convolution product into an integral, what you obtain is thus the general expression involving a Green function G(t,u)=g(t-u), where g(v) is zero for negative values of v.
The case where coefficients are constant is therefore much simpler than the general case: Where you had a two-variable integrator, you now have a single-variable one. Not only that, but the homogeneous solutions are well-known (if z is an eigenvalue of multiplicity n+1 for the matrix involved, the product of exp(zt) by any polynomial of degree n, or less, is a solution). In the important special case where all the eigenvalues are distinct, the determinants involved in the expression of G(t,u)=g(t-u) are essentially Vandermonde determinants, or Vandermonde cofactors (a Vandermonde determinant is a determinant where each column consists of the successive powers of a particular number). The expression is thus fairly easy to work out and may be put into the following simple form, involving the characteristic polynomial P for the equation (it's also the characteristic polynomial of the matrix we called A in the above). For any eigenvalue z, the derivative P'(z) is the product of the all the differences between that eigenvalue and each of the others (which is what Vandermonde expressions entail):
g(v) = exp(z1v) / P'(z1) + exp(z2v) / P'(z2) + ... + exp(znv) / P'(zn) if v>0
With this, x = g*b is indeed a special solution of our original equation f*x = b
The (unilateral) Laplace transform F = L( f ) of a function f is given by:
That's defined, for p > 0, whenever the integral makes sense. For example, the Laplace transform of a constant k is the function F such that F(p) = k/p.
Integrating by parts f' (t) exp(-pt) dt gives a simple relation between the respective Laplace transforms L( f') and L( f ) of f' and f :
L( f' ) = - f (0) + p L( f )
This may be iterated, starting with:
Oliver HeavisideThats the basis for the Operational Calculus, perfected in 1893 by Oliver Heaviside (1850-1925), to translate many practical systems of differential equations into algebraic ones. (Originally, Heaviside was interested in the transient solutions to the simple differential equations arising in electrical circuits).
Let's use capital letters to denote Laplace transforms of lowercase functions (W=L(w), Y=L(y), Z=L(z)). The differential system at hand translates into:
With other initial conditions, solutions may involve various linear combinations of no fewer than 5 different types of functions (namely: sin(t), cos(t), exp(-t), t and the constant 1), which would make a better showcase for Operational Calculus than this particularly simple example...
Below is a small table of Laplace transforms, which is sufficient to solve the above for any set of initial conditions, by reverse lookup :
1) Consider any nonmeasurable set E within the interval [0,1] (Zermelo's Axiom of Choice guarantees one such set exists) and define f(x) to be:
The function f is not Lebesgue-integrable, but its absolute value clearly is (|f(x)| is equal to 1 on [0,1] and 0 elsewhere).
That was for Lebesgue integration. For Riemann integration, you may construct a simpler example by letting the above E be the set of rationals between 0 and 1.
2) On the other hand, the function sin(x)/x is a simple example of a function which is Riemann-integrable over ]-¥,+¥[ (Riemann integration can be defined over an infinite interval, although it's not usually done in basic textbooks), whereas the absolute value |sin(x)/x| is not. Neither function is Lebesgue-integrable over ]-¥,+¥[, although both are over any finite interval.
The notation pertains to functional analysis. Let's explain:
If f (x) is the converging sum of all terms an xn (for some scalar sequence an ), f is called an analytic function [about zero] and it can be defined for some nonnumerical things that can be added, scaled or "exponentiated"...
The possibility of exponentiation to the power of a nonnegative integer reasonably requires the definition of some kind of multiplication with a neutral element (in order to define the zeroth power) but that multiplication need not be commutative or even associative. Power-associativity suffices (which is implied, in particular, by the alternativity obeyed by octonions). Here we merely focus on the multiplication of square matrices of finite sizes which corresponds to the composition of linear functions in a vector space of finitely many dimensions.
If M is a finite square matrix representing some linear operator (which we shall denote by the same symbol M for convenience) f (M) is defined as a power series of M. If there's a vector basis in which the operator M is diagonal, f (M) is diagonal in that same basis, with f (z) appearing on the diagonal of f (M) wherever z appears in the diagonal of M.
Now, the differential operator D is a linear operator like any other, whether it operates on a space of finitely many dimensions (for example, polynomials of degree 57 or less) or infinitely many dimensions (polynomials, formal series...). f (D) may thus be defined the same way. It's a formal definition which may or may not have a numerical counterpart, as the formal series involved may or may not converge. The same thing applies to any other differential operator, and this is how f (D) and f (D+a) are to be interpreted.
To prove that a linear relation holds when f appears homogeneously (as is the case here), it is enough to prove it for any n when f (x)=xn:
This completes a proof by induction for any f (x) = xn, which establishes the relation for any analytic function f, through summation of such elementary results (when such a summation makes sense).
Let's consider the class C of all functions whose (improper) integral from -¥ to +¥ is well-defined (or "convergent") at least as a Cauchy principal value. The following is trivially true for the case u(x) = x-a.
" f Î C , ò f (x) dx = ò f (u(x)) dx
What was discovered by Cauchy in 1823 (and rediscovered later by Schlömilch) is that this is also true in one nontrivial family of cases:
u(x) = x - k/x (for any constant k ≥ 0)
In 1983, Glasser pointed out that the above was more generally true for any finite sum of the following type:
u(x) = x - a - k1/(x-t1) - k2/(x-t2) - ... - kn/(x-tn)
We may interpret the sum as a Riemann sum whose limit defines an integral and find the following substitution also acceptable for any nonnegative distribution k (provided only that the convolution integral converges):
u(x) = x - a - ò k(t) (x-t)-1 dt
Cauchy-Schlömilch transformation (1823)
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Augustin Cauchy (1789-1857)
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Oscar Schlömilch (1823-1901)
Glasser's master theorem (1983)
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M. Lawrence Glasser
(b. 1933;
Ph.D. 1962;
Emeritus at Clarkson,
2005).
Ramanujan's master theorem
|
Srinivasa Ramanujan (1887-1920).
Big Integral Shortcut (16:50)
by Steve Chow (blackpenredpen, 2019年11月22日).
Richard Feynman (1918-1988) was instrumental in the popularization of this technique, which he had learned as a teenager from the 1926 textbook Advanced Calculus by Frederick S. Woods (1864-1950).
The general rule for differentiation under the integral is ultimately just a joint application the fundamental theorem of calculus and the chain rule, using partial derivatives. Sometimes dubbed Leibniz integral rule, this gives the expression of the derivative along a parameter p of an integral whose integrand and bounds of integrations may depend on p :
One important special case is when the lower bound (a) is constant and the upper bound of integration is simply b(p) = p :
The simplest special case is when both bounds (a and b) are constant:
Straight Leibniz Integral RuleRichard Feynman popularized the use of this formula to work out definite integrals which are otherwise difficult. The trick is to introduce an ad hoc parametric family of functions f (p,t) which includes the function f (t) whose definite integral we seek. E.g., f (t) = f (0,t).
The derivative of g is then obtained by differentiating under the integral sign, according to the above Leibniz integral rule. We're thus left with a differential equation for g whose full solution would yield the value of g (0) we're after. (If we need several successive differentations under the integral sign, we obtain a higher-order differential equation.)
With the above notations, f (t) = sin(t)/t and f (p,t) = e-pt sin(t)/t This yields ¶p f (p,t) = - e-pt sin(t). Therefore, by the Leibniz rule:
Thus g' (p) = -1 / (p2+1)
So, g (p) = C - Arctg p
The constant C is obtained from the fact that the integral which originally defined g (p) clearly tends to zero as p tends to infinity. Therefore:
g (p) = p/2 - Arctg p and g (0) = p/2
As f is an even function, its integral over the whole line is twice as big:
Dirichlet IntegralLet's build on this result with f (t) = sin2(t)/t2 and f (p,t) = sin2(pt)/t2 This yields ¶p f (p,t) = sin(2pt)/t. So, the Leibniz rule gives an integral equated to the above result by linear substitution (u = 2pt). Namely:
Therefore, g (p) = p p (since g (0) = 0) and g (1) = p , which is to say:
Leibniz's rule (and Feynman's trick)
by Zachary Lee
(Philosophical Math. 2017年07月22日).
Complexifying the Integral (9:22)
by Arthur Mattuck (MIT, Spring 2004).
Definite integral of sin(x)/x using Feynman's trick (22:43)
by Steve Chow (blackpenredpen, 2017年08月20日).
Feynman's Trick to Evaluate Integrals (3:22)
by Aditya Narayan Sharma (ANS Academy, 2017年07月10日).
Integration by Differentiating under the Integral Sign (9:03)
by Andrew Dotson (2018年05月11日).
These are due to Carl Johan Malmsten, but were rediscovered several times. The record was set straight recently (2012) by Iaroslav V. Blagouchine (b. 1979, PhD in 2001 and 2010). However, the simplest member of the family remains associated with the name of Ilan Vardi (1957-, PhD 1982) who rediscovered it in 1988.
[画像: Come back later, we're still working on this one... ]
Vardi's integral (Malmsten 1842, Vardi 1988)
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Carl Johan Malmsten (1814-1886)
Vardi integral (31:54) and
Malmsten integral (1:12:52)
by Dr. Peyam (2019).
