Appendix Special Functions Special functions are found to be of particular importance in mathematical analysis or mathematical physics or in other applications. This definition is possibly vague, since there is no consensus on a general definition of special functions, but there exists a common agreement on a large amount of them, like those reported in this appendix. Many special functions, and all those reported here, appear either as integrals of some elementary functions or as solutions of differential equations. The following description of some kinds of special functions is limited to those used in this book, without any claim of completeness, and the interested reader is invited to refer to specialised books like [1–3] and other that are specifically mentioned in the text, for more extensive treatments. A.1 The Gamma, Digamma and Beta Functions and Elliptic Integrals Although practically all special functions admit an integral representation, i.e. they can be written as the result of a definite integral, those reported in this section have a quite direct relationship with a definite integral. Beside the elliptic integrals,the Gamma function was for some time called Euler integral of second kind (named after the celebrated Swiss mathematician Leonhard Euler, 1707–1783); the Digamma function is the logarithmic derivative of the Gamma function while the Beta function was also called the Eulerintegraloffirstkind. A.1.1 The Gamma Function The Gamma function is a quite ubiquitous special function, since it appears in the most disparate areas of physics and mathematics, and often in mathematical relations among other special functions. The treatment of the Gamma function given in this © Springer Nature Switzerland AG 2021 371 G. E. Cossali and S. Tonini, Drop Heating and Evaporation: Analytical Solutions in Curvilinear Coordinate Systems, Mathematical Engineering, https://doi.org/10.1007/978-3-030-49274-8 372 Appendix: Special Functions appendix is deliberately brief and the interested reader can refer, for a more complete treatment, to specialised books like [4, 5]. The first representation of the Gamma function was given by Daniel Bernoulli (a Swiss mathematician and physicist, 1700–1782) and Leonhard Euler in two letters written independently to Goldbach (Christian Goldbach, a German mathematician, 1690–1764) in October 1729 [6]; at that time Daniel Bernoulli and Leonhard Euler were working together in St. Petersburg. The Gamma function can be seen as an extension of the factorial to complex and real number arguments, and it was the research on this problem that led Euler to suggest the first integral representation ∞ (z) = e−t t z−1dt; Re {z} > 0(A.1) 0 The notation (x) is due to Legendre [7](seealso[8]), and the relation with the factorial is (n + 1) = n! (A.2) Some special values of (z) are the following 1 (1) = 1; = π1/2; (1) = γ (A.3) 2 where n 1 γ = lim − ln (n) = 0.5772156649.... (A.4) n→∞ k k=1 is the Euler’s constant. The Gamma function can be defined for any complex number z (see [3] for a detailed analysis) with the exception of negative integer numbers. The Gamma function has no zeros in the complex plane [3] and Fig. A.1 shows (z) for real values of z. The Gamma function satisfies an important recurrence relation (z + 1) = z (z) (A.5) that can be found directly from the integral representation, in fact ∞ ∞ ( + ) = −t z =− −t z ∞ + −t z−1 z 1 e t dt e t 0 z e t dt (A.6) 0 0 and since the first term in the most RHS is nil, Eq. (A.5) is proven. Equation (A.5) is useful to find values of the function without performing other integrations, for example Appendix: Special Functions 373 Fig. A.1 Gamma function (z) for real values of z 3 3 (3/2) = (1/2) = π1/2 2 2 5 15 (5/2) = (3/2) = π1/2 (A.7) 2 4 ... There are other two important relations satisfied by (z) π (z) (z − 1) = ; z = 0, ±1, ±2,... (A.8) sin (zπ) 22z−1 (2z) = (z) (z + 1/2) ; 2z = 0, −1, −2,... (A.9) π1/2 which can again be obtained directly from Eq. (A.1), see [3] for a derivation. A.1.2 The Digamma Function Another important function related to the Gamma function is its logarithmic deriva- tive, or d ln (z) (z) ψ (z) = = (A.10) dz (z) 374 Appendix: Special Functions Fig. A.2 Digamma function ψ (z) for real values of z called Digamma function or Psi function. Figure A.2 shows its graph for real values of z. The analogous of the relations (A.5), (A.8) and (A.9)are 1 ψ (z + 1) − ψ (z) = (A.11) z π ψ (1 − z) − ψ (z) = ; z = 0, ±1,... (A.12) tan (zπ) 1 ψ (2z) − [ψ (z) + ψ (z + 1/2)] = ln 2; 2z = 0, −1, −2,... (A.13) 2 and some special values can be found applying the definition or these relations, for example (1) ψ (1) = =−γ (A.14) (1) and from (A.12) ψ (1/2) = ψ (1) − 2ln2=−γ − 2 ln 2 (A.15) Appendix: Special Functions 375 Values of ψ (z) for negative argument can be calculated from those with positive argument by (A.12) π ψ (−z) = ψ (1 + z) + (A.16) tan (zπ) For many different integral representations of ψ (z) see [1]. A.1.3 The Beta Function A third function that appears relatively often in the theory of special functions and in many applications is the so-called Beta function, a name that was given to this function, also called Euler integral of first kind, by Binet (a French mathematician, 1786–1856) on 1839 [9] in a paper (see also [8]), where he proposed the symbol B, which is the Greek capital letter beta. The integral representation of the Beta function is 1 B (a, b) = ta−1 (1 − t)b−1 dt (A.17) 0 and it bears a simple relation to the Gamma function (a) (b) B (a, b) = (A.18) (a + b) It satisfy various identities (see [1]), the following π B (a, 1 − a) = (A.19) sin (πa) setting a = 1/2, in conjunction with Eq. (A.18) yields the result π 2 (1/2) π = = B (1/2, 1/2) = = 2 (1/2) (A.20) sin (π/2) (1) i.e. (1/2) = π1/2. Equation (A.18) suggest also a relation with binomial coefficient for a, b integers m m! 1 (m + 1)! 1 (m + 2) = = = (A.21) n n! (m − n)! m + 1 n! (m − n)! m + 1 (n + 1) (m − n + 1) = 1 (m + 1) B (n + 1, m − n + 1) Due to Eq. (A.18) all the properties of this function can be derived from those of (z). 376 Appendix: Special Functions A.1.4 The Elliptic Integrals An integral of the form I = F (t,w) dt (A.22) where F is a rational function of w and t (i.e. it can be written as the ratio between 2 two polynomials in w and t) and w is a polynomial of third or fourth degree in t (i.e. w2 = 4 k = = k=0 ak t where a4 0ora3 0 or both) is said to be an elliptic integral. The name is due to the fact that these integrals were found when trying to calculate the length of an arc of ellipse. The first to use elliptic integral was probably Fagnano (Giulio Carlo Fagnano, an Italian mathematician, 1682–1766) while studying the rectification of the lemniscate [10], while the foundation of the elliptic integrals and elliptic functions theory is due to Euler. The most important elliptic integrals are those called incomplete (Legendre) ellip- tic integral of first, F (φ, t), second, E (φ, t) and third, φ, α2, t kind, defined as φ dx sin(φ) dt F (φ, t) = = √ √ (A.23) 2 0 1 − k2 sin (x) 0 1 − t2 1 − k2t2 √ φ sin(φ) 1 − k2t2 E (φ, t) = 1 − k2 sin2 (x)dx = √ dt (A.24) 0 0 1 − t2 φ 1 φ, α2, t = dx 2 2 0 1 − k2 sin (x) 1 − α2 sin (x) sin(φ) dt = √ √ (A.25) 0 1 − t2 1 − k2t2 1 − α2t2 where φ is the amplitude (or argument), k is the elliptic modulus (or eccentricity) while α2 in the third one is called characteristic. Sometimes the parameter m = k2 is used instead of the modulus, or also the complementary modulus k defined by k2 + k2 = 1. The complete elliptic integrals are instead π K (k) = F , t first kind (A.26) 2 π E (k) = E , t second kind (A.27) 2 π α2, t = , α2, t third kind (A.28) 2 Some special values are the following Appendix: Special Functions 377 π F (φ, k) F (0, k) = 0; F (φ, 0) = φ; F , 1 =∞; lim = 1 (A.29) 2 φ→0 φ π E (φ, k) E (0, k) = 0; E (φ, 0) = φ; E , 1 = 1; lim = 1(A.30) 2 φ→0 φ (φ, 0, t) = F (φ, t) (A.31) π K (0) = E (0) = ; K (1) =∞; E (1) = 1(A.32) 2 The complete integrals satisfy some connection formulas and identities like π = E k K (k) − K (k) K (k) + E (k) K k (A.33) 2 ⎧ ⎨ 2 1 kK (k) − ikK k if Im k > 0 = K (A.34) k ⎩ kK (k) + ikK k if Im k2 < 0 ⎧ ⎨ 1 2 2 2 1 E (k) + iE k − k K (k) − ik K k if Im k > 0 E = k (A.35) k ⎩ 1 ( ) − − 2 ( ) + 2 2 < k E k iE k k K k ik K k if Im k 0 and many others (see [1]).