Special Functions
This worksheet gives definitions, properties, and graphs of some of the special mathematical functions available in Maple.
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The Airy Wave Functions
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The Airy Ai and Bi functions, AiryAi(z) and AiryBi(z) respectively, solve the differential equation:
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Floating-point Evaluation
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| (1.1.1) |
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Differentiation
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The derivatives of the Airy functions are denoted by AiryAi(1, z) and AiryBi(1, z):
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| (1.2.1) |
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| (1.2.2) |
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Graphing
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The Anger and Weber Functions
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The Anger J function, AngerJ(nu, z), solves the inhomogeneous Bessel equation:
and the Weber E function, WeberE(nu, z), solves the inhomogeneous Bessel equation:
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Floating-point Evaluation
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| (2.1.1) |
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| (2.1.2) |
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| (2.1.3) |
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| (2.1.4) |
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Differentiation
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The derivatives with respect to the variable can be expressed in terms of algebraic expressions in trigonometric functions and the Anger and Weber functions themselves with shifted parameters:
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| (2.2.1) |
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| (2.2.2) |
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The Bessel Functions
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The Bessel J and Y functions of the first kind, BesselJ(nu, z) and BesselY(nu,z), satisfy the differential equation:
The Bessel I and K functions of the second kind, BesselI(nu,z) and BesselK(nu,z), satisfy the modified differential equation:
The Hankel and functions, HankelH1(nu,z) and HankelH2(nu,z) are related to the Bessel functions of the first kind by:
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Floating-point Evaluation
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| (3.1.1) |
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Differentiation
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The derivatives of the Bessel and Hankel functions may be expressed as algebraic expressions in themselves with shifted parameters:
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| (3.2.1) |
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| (3.2.2) |
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| (3.2.3) |
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| (3.2.4) |
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| (3.2.5) |
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| (3.2.6) |
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The Kummer (Confluent Hypergeometric) Functions
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The Kummer M and U functions, KummerM(mu, nu, z) and KummerU(mu, nu, z), are solutions of the differential equation
which can be written as:
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Floating-point Evaluation
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Asymptotic approximations:
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| (4.1.1) |
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| (4.1.2) |
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| (4.1.3) |
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| (4.1.4) |
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Differentiation
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The derivatives of the Kummer functions may be expressed as algebraic expressions in themselves with shifted parameters:
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| (4.2.1) |
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| (4.2.2) |
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Graphing
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The Parabolic Cylinder Functions
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The cylinder U and V functions, CylinderU(a, z) and CylinderV(a,z) respectively, are solutions to the differential equation:
The Whittaker's parabolic D function, CylinderD(a,z), is related to the above by the following expression
which are equal when .
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Floating-point Evaluation
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| (5.1.1) |
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| (5.1.2) |
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Differentiation
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The derivatives of the cylinder functions can be written as simple algebraic expressions of themselves with modified parameters:
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| (5.2.1) |
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| (5.2.2) |
The derivative of the Whittaker parabolic D function is cleaner than the derivative of the cylinder U function:
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| (5.2.3) |
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Graphing
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The Elliptic Functions
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The incomplete elliptic F function of the first kind, EllipticF(z,k), and complete elliptic K and CK functions of the first kind, EllipticK(k) and EllipticCK(k), are defined as:
The incomplete elliptic E function of the second kind, EllipticE(z,k), and complete elliptic E and CE functions of the second kind, EllipticE(k) and EllipticCE(k), are defined as:
The incomplete elliptic Pi function of the third kind, EllipticPi(z,nu,k), and complete elliptic Pi and CPi functions of the second kind, EllipticPi(nu,k) and EllipticCPi(nu,k), are defined as:
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Floating-point Evaluation
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| (6.1.1) |
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| (6.1.2) |
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| (6.1.3) |
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Differentiation
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The derivatives of the elliptic functions of the first and second kinds are in the form of algebraic expressions in themselves:
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| (6.2.1) |
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| (6.2.2) |
The derivatives of the elliptic functions of the third kind are algebraic expressions in all three kinds of elliptic functions:
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| (6.2.3) |
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| (6.2.4) |
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Graphing
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The Jacobi Elliptic Functions
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The Jacobi sn, cn, dn, ..., dc functions, JacobiSN(z,k), JacobiCN(z,k), JacobiDN(z,k), ..., JacobiDC(z,k) are defined in terms of the Jacobi am amplitude function JacobiAM(z, k):
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Floating-point Evaluation
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| (7.1.1) |
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| (7.1.2) |
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| (7.1.3) |
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| (7.1.4) |
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Differentiation
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The derivatives of the Jacobi elliptic functions are expressions in other Jacobi elliptic functions:
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| (7.2.1) |
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| (7.2.2) |
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Graphing
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The Jacobi Theta Functions
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The Jacobi , , , and functions JacobiTheta1(z, q), JacobiTheta2(z, q), JacobiTheta3(z, q), and JacobiTheta4(z, q) are defined as:
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Floating-point Evaluation
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| (8.1.1) |
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| (8.1.2) |
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| (8.1.3) |
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Differentiation
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The derivatives of the Jacobi theta functions are less elegant than other special functions:
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| (8.2.1) |
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| (8.2.2) |
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Graphing
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The Riemann and Hurwitz Zeta Functions
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The Riemann function Zeta may be defined as
or
where the product is over all primes p. The Hurwitz Zeta function is defined as:
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Floating-point Evaluation
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| (9.1.1) |
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| (9.1.2) |
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| (9.1.3) |
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Differentiation
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The derivative of the Zeta function is denoted as Zeta(1,z):
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| (9.2.1) |
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| (9.2.2) |
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Graphing
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The Jacobi Zeta Function
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The Jacobi Zeta function is defined by
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Floating-point Evaluation
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| (10.1.1) |
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Differentiation
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The derivatives of the Jacobi Zeta function can be written in terms of elliptic functions:
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| (10.2.1) |
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Graphing
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The Kelvin Functions
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The Kelvin Ber and Bei functions, KelvinBer(nu, z) and KelvinBei(nu,z), are defined by the following equations:
The Kelvin Ker and Kei functions, KelvinKer(nu,z) and KelvinKei(nu,z), are defined by the following equations:
The Kelvin Her and Hei functions, KelvinHer(nu,z) and KelvinHei(nu,z), are defined by the following equations:
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Floating-point Evaluation
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| (11.1.1) |
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| (11.1.2) |
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Differentiation
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The derivatives of the Kelvin functions can be written in terms of algebraic expressions in themselves with shifted parameters:
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| (11.2.1) |
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| (11.2.2) |
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| (11.2.3) |
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The Legendre Functions
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The Legendre P and Q functions of the first and second kinds, LegendreP(nu, z) and LegendreQ(nu, z), solve the differential equation:
The associated Legendre P and Q functions of the first and second kinds, LegendreP(nu, mu, z) and LegendreQ(nu, mu, z), solve the differential equation:
The associated case simplifies to the regular case when .
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Floating-point Evaluation
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| (12.1.1) |
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| (12.1.2) |
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Differentiation
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The derivative of the Legendre functions can be written as algebraic expressions in themselves with shifted parameters:
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| (12.2.1) |
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| (12.2.2) |
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| (12.2.3) |
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| (12.2.4) |
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Graphing
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The Lommel Functions
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The Lommel s and S functions, LommelS1(mu, nu, z) and LommelS2(nu, z), are defined by the following equations:
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Floating-point Evaluation
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The difference between the Anger J and Bessel J functions is approximately equal to the second expression in terms of Lommel S functions:
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| (13.1.1) |
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| (13.1.2) |
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| (13.1.3) |
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| (13.1.4) |
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Differentiation
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The derivative of the Lommel functions can be written in terms of algebraic expressions in themselves with shifted parameters:
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| (13.2.1) |
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| (13.2.2) |
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Graphing
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The Weierstrass Functions
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The Weierstrass , , , and functions, WeierstrassP(z, g2, g3), WeierstrassPPrime(z, g2, g3), WeierstrassZeta(z, g2, g3), and WeierstrassSigma(z, g2, g3) may be defined as follows:
and,
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Floating-point Evaluation
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| (14.1.1) |
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| (14.1.2) |
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| (14.1.3) |
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| (14.1.4) |
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Differentiation
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The derivatives of the various Weierstrass functions can be written in terms of simple algebraic expressions of other Weierstrass functions:
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| (14.2.1) |
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| (14.2.2) |
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| (14.2.3) |
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| (14.2.4) |
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Graphing
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The Whittaker Functions
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The Whittaker M and W functions, WhittakerM(mu, nu, z) and WhittakerW(mu, nu, z), solve the differential equation:
They are related to each other by:
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Floating-point Evaluation
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| (15.1.1) |
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| (15.1.2) |
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Differentiation
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The derivatives of the Whittaker functions can be written as algebraic expressions in themselves with a shifted first parameter:
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| (15.2.1) |
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| (15.2.2) |
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For more information, consult the help pages for the individual functions.
Return to Index for Example Worksheets
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