relation to modulus and phase

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21: 1.9 Calculus of a Complex Variable

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Modulus and Phase

… ►The principal value of $\mathrm{ph}z$ corresponds to $n=0$, that is, $-\pi \le \mathrm{ph}z\le \pi$. …(However, if we require a principal value to be single-valued, then we can restrict .) … ►Equations (1.9.18) and (1.9.20) hold for general values of the phases, but not necessarily for the principal values. … ►A series ${\sum }_{n=0}^{\mathrm{\infty }}{z}_n$ converges (diverges) absolutely when ($>1$), or when ($>1$). …

22: 12.14 The Function $W(a,x)$

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§12.14(vii) Relations to Other Functions

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Bessel Functions

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Confluent Hypergeometric Functions

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§12.14(x) Modulus and Phase Functions

… ►For properties of the modulus and phase functions, including differential equations and asymptotic expansions for large $x$, see Miller (1955, pp. 87–88). …

23: 1.12 Continued Fractions

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Recurrence Relations

… ►A continued fraction converges if the convergents $C_n$ tend to a finite limit as $n\to \mathrm{\infty }$. … ►and the even and odd parts of the continued fraction converge to finite values. …In this case $\left|\mathrm{ph}C\right|\le \frac{1}{2}\pi$. … ►For analytical and numerical applications of continued fractions to special functions see §3.10. …

24: 7.18 Repeated Integrals of the Complementary Error Function

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§7.18(iv) Relations to Other Functions

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Hermite Polynomials

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Confluent Hypergeometric Functions

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Parabolic Cylinder Functions

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Probability Functions

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25: 22.20 Methods of Computation

… ►A powerful way of computing the twelve Jacobian elliptic functions for real or complex values of both the argument $z$ and the modulus $k$ is to use the definitions in terms of theta functions given in §22.2, obtaining the theta functions via methods described in §20.14. … ►for $n\ge 1$, where the square root is chosen so that $\mathrm{ph}{b}_n=\frac{1}{2}(\mathrm{ph}{a}_{n-1}+\mathrm{ph}{b}_{n-1})$, where $\mathrm{ph}{a}_{n-1}$ and $\mathrm{ph}{b}_{n-1}$ are chosen so that their difference is numerically less than $\pi$. … ►

§22.20(vi) Related Functions

… ►Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute $\mathrm{am}(x,k)$. … ►For additional information on methods of computation for the Jacobi and related functions, see the introductory sections in the following books: Lawden (1989), Curtis (1964b), Milne-Thomson (1950), and Spenceley and Spenceley (1947). …

26: 13.2 Definitions and Basic Properties

… ►It can be regarded as the limiting form of the hypergeometric differential equation (§15.10(i)) that is obtained on replacing $z$ by $z/b$, letting $b\to \mathrm{\infty }$, and subsequently replacing the symbol $c$ by $b$. … ►Although $M(a,b,z)$ does not exist when $b=-n$, $n=0,1,2,\mathrm{\dots }$, many formulas containing $M(a,b,z)$ continue to apply in their limiting form. … ►Unless specified otherwise, however, $U(a,b,z)$ is assumed to have its principal value. ►

§13.2(iii) Limiting Forms as $z\to 0$

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§13.2(iv) Limiting Forms as $z\to \mathrm{\infty }$

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27: 1.10 Functions of a Complex Variable

… ►An isolated singularity $z_0$ is always removable when ${lim}_{z\to z_0}f(z)$ exists, for example $(\sin z)/z$ at $z=0$. … ►

Phase (or Argument) Principle

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§1.10(v) Maximum-Modulus Principle

… ►It should be noted that different branches of ${(w-w_0)}^{1/\mu }$ used in forming ${(w-w_0)}^{n/\mu }$ in (1.10.16) give rise to different solutions of (1.10.12). … ►

§1.10(x) Infinite Partial Fractions

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28: 2.1 Definitions and Elementary Properties

… ►In (2.1.5) $ℝ$ can be replaced by any fixed ray in the sector , or by the whole of the sector $|\mathrm{ph}x|\le \frac{1}{2}\pi -\delta$. …But (2.1.5) does not hold as $x\to \mathrm{\infty }$ in (for example, set $x=1+\mathrm{i}t$ and let $t\to \pm \mathrm{\infty }$.) … ►Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. … ►Condition (2.1.13) is equivalent to … ►If the set $𝐗$ in §2.1(iii) is a closed sector $\alpha \le \mathrm{ph}x\le \beta$, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\mathrm{ph}x\in [\alpha ,\beta ]$ as $|x|\to \mathrm{\infty }$. …

29: 36.5 Stokes Sets

… ►The Stokes set consists of the rays $\mathrm{ph}x=\pm 2\pi /3$ in the complex $x$-plane. … ►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: … ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►This consists of three separate cusp-edged sheets connected to the cusp-edged sheets of the bifurcation set, and related by rotation about the $z$-axis by $2\pi /3$. … ►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …

30: 19.2 Definitions

… ►The cases with $\varphi =\pi /2$ are the complete integrals: … ►with a branch point at $k=0$ and principal branch $|\mathrm{ph}k|\le \pi$. … ►If , then $k_c$ is pure imaginary. ►Lastly, corresponding to Legendre’s incomplete integral of the third kind we have … ►

§19.2(iv) A Related Function: $R_C(x,y)$

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