"Foundations of Black Hole Accretion Disk Theory"
Marek A. Abramowicz and P. Chris Fragile 
1 Introduction
2 Three Destinations in Kerr’s Strong Gravity
2.1 The event horizon
2.2 The ergosphere
2.3 ISCO: the orbit of marginal stability
2.4 The Paczyński–Wiita potential
2.5 Summary: characteristic radii and frequencies
3 Matter Description: General Principles
3.1 The fluid part
3.2 The stress part
3.3 The Maxwell part
3.4 The radiation part
4 Thick Disks, Polish Doughnuts, & Magnetized Tori
4.1 Polish doughnuts
4.2 Magnetized Tori
5 Thin Disks
5.1 Equations in the Kerr geometry
5.2 The eigenvalue problem
5.3 Solutions: Shakura–Sunyaev & Novikov–Thorne
6 Slim Disks
7 Advection-Dominated Accretion Flows (ADAFs)
8 Stability
8.1 Hydrodynamic stability
8.2 Magneto-rotational instability (MRI)
8.3 Thermal and viscous instability
9 Oscillations
9.1 Dynamical oscillations of thick disks
9.2 Diskoseismology: oscillations of thin disks
10 Relativistic Jets
11 Numerical Simulations
11.1 Numerical techniques
11.2 Matter description in simulations
11.3 Polish doughnuts (thick) disks in simulations
11.4 Novikov–Thorne (thin) disks in simulations
11.5 ADAFs in simulations
11.6 Oscillations in simulations
11.7 Jets in simulations
11.8 Highly magnetized accretion in simulations
12 Selected Astrophysical Applications
12.1 Measurements of black-hole mass and spin
12.2 Black hole vs. neutron star accretion disks
12.3 Black-hole accretion disk spectral states
12.4 Quasi-Periodic Oscillations (QPOs)
12.5 The case of Sgr A*
13 Concluding Remarks

2 Three Destinations in Kerr’s Strong Gravity

In this section, we briefly describe the three destinations within Kerr’s strong gravity that are most relevant to black hole accretion disk theory:
  1. Event Horizon: That radius inside of which escape from the black hole is not possible;
  2. Ergosphere: That radius inside of which negative energy states are possible (giving rise to the potentiality of tapping the energy of the black hole).
  3. Innermost Stable Circular Orbit (ISCO): That radius inside of which free circular orbital motion is not possible;

Our principal question is: Could accretion disk theory unambiguously prove the existence of the event horizon, ergosphere, and ISCO using currently available or future observations?

In realistic astrophysical situations involving astrophysical black holes (in particular quasars and microquasars), the black hole itself is uncharged, and the gravity of accretion disk is practically negligible. This means that the spacetime metric gμν is given by the Kerr metric, determined by two parameters: total mass M ∗ and total angular momentum J∗. It is convenient to rescale them by

GM M = ---2∗ (2a ) c a = -J∗-, (2b ) M ∗c

such that both M and a are measured in units of length.

In the standard spherical Boyer–Lindquist coordinates the Kerr metric takes the form [31Jump To The Next Citation Point],

( 2M r) (r2 + a2 )2 − a2Δ sin2𝜃 gtt = − 1 − ----- , gtt = − ----------------------, ϱ2 ϱ2Δ 2M ar sin2 𝜃 2M ar gtϕ = − -----2-----, gtϕ = − --2---, ( )ϱ ϱ Δ 2 2 2M--a2rsin2𝜃- 2 ϕϕ Δ-−-a2-sin2-𝜃- g ϕϕ = r + a + ϱ2 sin 𝜃, g = Δ ϱ2 sin2 𝜃 , 2 g = ϱ--, g = ϱ2, grr = -Δ-, g𝜃𝜃 = -1-, (3 ) rr Δ 𝜃𝜃 ϱ2 ϱ2
where 2 2 Δ = r − 2M r + a and 2 2 2 2 ϱ = r + a cos 𝜃.

The Kerr metric depends neither on time t, nor on the azimuthal angle ϕ around the symmetry axis. These two symmetries can be expressed in a coordinate independent way by the two commuting Killing vectors ημ = δ μt and ξμ = δμ ϕ,

∇ μηλ + ∇ λη μ = 0, ∇ μξλ + ∇ λξμ = 0, (4a ) μ ν μ ν η ∇ μξ = ξ ∇ μη . (4b )

Here ∇ μ denotes the covariant derivative,

ν ν 1 νκ ∇ μηλ = ∂μηλ − Γ μλην, Γ μλ = 2g (∂ λgμκ + ∂μgλκ − ∂κgλμ), (5 )
and μ ∂μ = ∂∕∂x denotes the standard partial derivative. Formulae for the Kerr metric (3View Equation) and all its non-zero Christoffell symbols Γ νμλ (5View Equation), are available from [305].

In Boyer–Lindquist coordinates the t and ϕ components of the Kerr metric can be expressed as scalar products of the Killing vectors,

g = ημη , g = ημξ , g = ξμξ . (6 ) tt μ tϕ μ ϕϕ μ
From the Killing vectors one can also define the following constants of motion for a particle or photon with four-momentum pμ
energy: ℰ ≡ − η μp = − p, (7a ) μ t angular momentum: ℒ ≡ ξμpμ = pϕ, (7b ) ℒ pϕ uϕ specific angular momentum: ℓ ≡ -- = − ---= − --, (7c ) ℰ pt ut μν 2 -p2ϕ--- Carter constant: 𝒞 = K pμpν = (p𝜃) + sin2 𝜃. (7d )

The Carter constant 𝒞 is connected to the Killing tensor K μν, which exists in the Kerr metric. Killing tensors obey,

1- 6 [∇ κK μν + ∇ μK νκ + ∇ νK κμ + ∇ κK νμ + ∇νK μκ + ∇ μK κν] = 0. (8 )
The other coordinates often used in black hole accretion disk research are the Kerr–Schild coordinates, in which the metric takes the form,
g = η + f k k , η = diag(− 1, 1,1,1) μν μν μ ν μν -2M--r3-- f = r4 + a2z2 , k = 1, t rx + ay kx = -------, r2 + a2 k = ry-−-ax, y r2 + a2 z- kz = r, (9 )
where kν = (kt,kx,ky,kz) is a unit vector, and r is given implicitly by the condition,
x2 +-y2- z2- r2 + a2 + r2 = 1. (10 )
Note that we have given the Kerr–Schild metric in its Cartesian form to prevent confusion with the spherical-polar Boyer–Lindquist coordinates. In keeping with this, unless specifically stated otherwise, the indices {t,ϕ,r,𝜃} will always refer to the Boyer–Lindquist coordinates in this review.

2.1 The event horizon

The mathematically precise, general, definition of the event horizon involves topological considerations [207]. Here, we give a definition which is less general, but in the specific case of the Kerr geometry is fully equivalent.

The Boyer–Lindquist coordinates split the Kerr spacetime into a “time” coordinate t and a three-dimensional “space,” defined as t = const hypersurfaces. This split may be done in a coordinate independent way, based on the Killing vectors which exist in the Kerr spacetime. Indeed, the family of non-geodesic observers N μ with trajectories orthogonal to a family of 3-D spaces t = const is defined as,

μ −Φ μ μ μ μ ZAMO: N ≡ e η&tidle; , &tidle;η = η + ω ξ , (11a ) ηνξν- gtϕ-- frame dragging:ω ≡ − ξμξ = − g . (11b ) μ ϕϕ

They are called zero-angular-momentum-observers (ZAMO), because for them, the angular momentum defined by (7b) is zero, ℒ = N ϕ = 0. The ZAMO observers provide the standard of rest in the 3-D space: objects motionless with respect to the ZAMO frame of reference occupy fixed positions in space.

We can also define a gravitational potential in the ZAMO frame:

[ ] 1 ξμξμ 1 tt potential: Φ ≡ − --ln --ν---2----ν-----μ---- = − --ln |g |. (12 ) 2 (η ξν) − (η η ν)(ξ ξμ ) 2
The primary reason to call Φ the gravitational potential is that, in Newton’s theory, the observer who stays still in space experiences an acceleration due to “gravity” gμ, which equals the gradient of the gravitational potential. In the Kerr spacetime it is,
ν gμ = (aμ)ZAMO ≡ N ∇ νN μ = ∇ μΦ. (13 )
From (11a) one sees that at the surface 1∕gtt = 0, the vector &tidle;ημ is null, &tidle;ημη&tidle;μ = 0. Therefore, the ZAMO observers who provide the standard of rest, move on that surface with the speed of light. In order to stand still in this location, one must move radially out with the speed of light.11 As it is clear from (3View Equation), tt 1∕g = 0 is equivalent to Δ = 0. The last equation has a double solution,
( ∘ -------) a-- r = rH = rG 1 ± 1 − M . (14 )
Note, that for r < r+ the ZAMO “observers” are spacelike: standing still at a given radial location implies moving along a spacelike trajectory – i.e., faster than light. All trajectories that move radially out are also spacelike. Thus, the outer root r = r+ of Eq. (14View Equation) defines the Kerr black hole event horizon: a null surface that surrounds a region from which nothing may escape. Outside the outer horizon (i.e., for r > r+) the normalization of μ N is non-singular, and therefore the gravitational potential (12View Equation) is a non-singular, well-defined quantity.

2.1.1 Detecting the event horizon

One may think of two general classes of astrophysical observations that could provide evidence for a black hole horizon. Arguments in the first class are indirect; they are based on estimating a dimensionless “compactness parameter”

----(size-of the-object)---- ℭ = (G∕c2)(mass of the object). (15 )
Arguments in the second class are more direct. They are based (in principle) on showing that some amount of radiation emitted by the source is lost inside the horizon.

Evidence based on estimating the compactness parameter: A source for which observations indicate ℭ ≈ 1 may be suspected of having an event horizon. Values ℭ ≈ 1 have indeed been found in several astronomical sources. In order to know ℭ, one must know mass and size of the source. The mass measurement is usually a direct one, because it may be based on an application of Kepler’s laws. In a few cases the mass measurement is remarkably accurate. For example, in the case of Sgr A*, the supermassive black hole in the center of our Galaxy, the mass is measured to be 6 M = (4.3 ± 0.5) × 10 M ⊙ [111Jump To The Next Citation Point].

Until recently, estimates of size were always indirect, and generally not accurate. They are usually based on time variability or spectral considerations. For the former, the measurement rests on the logic that if the shortest observed variability time-scale is Δt, then the size of the source cannot be larger than R = cΔt. For the latter, the argument goes like this: If the total radiative power L and the radiative flux F can be independently measured for a black-body source, then its size can be estimated from 2 L = 4πR F. Keep in mind that one must know the distance to the source in order to measure L. The flux can be estimated from F = aT 4, where T is the temperature corresponding to the peak in the observed intensity versus frequency electromagnetic spectrum.

It is hoped that in the near future, the next generation of high-tech radio telescopes will be able to measure directly the size of “the light circle”, which is uniquely related to the horizon size (see Figure 2View Image). For Sgr A*, at a distance of 8.28 ± 0.44 kpc [111], the event horizon corresponds to an angular size of ∼ 10 μas in the sky, making it an ideal target for near-future microarcsecond very long base interferometric techniques [77, 83]. Here the plan is to measure the black hole “shadow” or “silhouette.”

View Image

Figure 2: Silhouettes of Sgr A* calculated for four optically thin accretion structures, characterized by very different physical conditions. The display is intentionally reversed in black-and-white and saturated in order to better show the less luminous parts. Although “dirty astrophysics” makes the most prominent differences, effects of the “pure strong gravity” are also seen in the form of “the light circle”, a tiny almost circular feature at the center. Its shape and size depends only on the black hole mass and spin. Image reproduced by permission from [297], copyright by ESO.

Evidence based on the “no escape” argument: For accretion onto an object with a physical surface (such as a star), 100% of the gravitational binding energy released by accretion must be radiated away. This does not apply for a black hole since the event horizon allows for the energy to be advected into the hole without being radiated. This may allow for a black hole with an event horizon to be distinguished from another, similar-mass object with a surface, such as a neutron star. This argument was first developed by Narayan and collaborators [215Jump To The Next Citation Point, 216, 214]; we describe it in more detail in Section 12.2.

2.2 The ergosphere

In Newtonian gravity, angular momentum ℓ and angular velocity Ω are related by the formula ℓ = r2Ω, and therefore there is no ambiguity in defining a non-rotating frame as Ω = 0 = ℓ. However, in the Kerr geometry ℓ ∝ (Ω − ω ), where ω = − gtϕ∕gϕϕ is the angular velocity of the frame dragging induced by the Lense–Thirring effect. Therefore, Ω = 0 does not imply ℓ = 0. This leads to two different standards of “rotational rest”: the Zero Angular Velocity Observer (ZAVO) and the Zero Angular Momentum Observer (ZAMO),

μ ν −1∕2 μ ZAVO frame (Ω = 0): n = (− η ην) η , (16 ) ZAMO frame (ℓ = 0): N μ = eΦ(ημ + ω ξμ). (17 )
These two frames rotate with respect to each other with the frame-dragging angular velocity ω = − gtϕ∕gϕϕ.

The ZAMO frame defines a local standard of rest with respect to the local compass of inertia: a gyroscope stationary in the ZAMO frame does not precess. Considering its kinematic invariants,12 one sees that the ZAMO frame is non-inertial (aμ ⁄= 0), non-rigid (σ μν ⁄= 0, Θ = 0), and surface-forming (ω = 0 μν). The ZAMO vectors &tidle;ημ and N μ are time-like everywhere outside the horizon, i.e., outside the surface tt 1∕g = 0. This means that the energy of a particle or photon with a four-momentum pμ measured by the ZAMO is positive, μ EZAMO ≡ N pμ > 0.

The ZAVO frame defines a global standard of rest with respect to distant stars: a telescope that points to a fixed star does not rotate in the ZAVO frame. Considering its kinematic invariants one sees that the ZAVO frame is non-inertial (aμ ⁄= 0), rigid (σμν = 0, Θ = 0), and not surface-forming (ω μν ⁄= 0). At infinity, i.e., for r → ∞, it is ν (η η ν) = gtt → − 1, and therefore μ μ n → η. For this reason, η μ is called the stationary observer at infinity. The ZAVO vectors ημ and n μ are timelike outside the region surrounded by the surface gtt = 0, called the ergosphere. Inside the ergosphere ημ and nμ are spacelike. This means that inside the ergosphere, the conserved energy of a particle (i.e., the energy measured “at infinity”), as defined by (7a), may be negative.

Penrose [242Jump To The Next Citation Point] considered a process in which, inside the ergosphere, a particle with energy E ∞ > 0 decays into two particles with energies E∞ > 0 + and E∞ = − |E ∞ | < 0 − −. The particle with positive energy escapes to infinity, and the particle with the negative energy gets absorbed by the black hole. Then, because ∞ ∞ ∞ E + = E − E− = E + |E − | > E, one gets a net gain of positive energy at infinity. The source of energy in this Penrose process is the rotational energy of the black hole. Indeed, the angular momentum absorbed by the black hole, J∞ = piξi is necessarily negative (in the sense that J∞ ωH < 0), which follows from

E ≡ − p eΦ(ηi + ω ξi) = eΦ(E ∞ − ω J ∞) > 0, ZAMO i H − H

and thus

ω J ∞ < E ∞ < 0. (18 ) H −
A more complete presentation of the Penrose process is made in [310]. At this time it appears the most likely realization of the Penrose process would be the Blandford–Znajek mechanism [49Jump To The Next Citation Point] for launching jets from quasars and microquasars. Observations suggest [255, 252Jump To The Next Citation Point], and simulations confirm [303Jump To The Next Citation Point, 304Jump To The Next Citation Point], that through this mechanism it is possible to extract more energy from the system than is being delivered by accretion. We discuss jets and the Blandford-Znajek mechanism more in Sections 10 and 11.7.

2.3 ISCO: the orbit of marginal stability

Particles (with velocity normalization uμuμ = − 1) and photons (with velocity normalization uμu μ = 0) move freely on “geodesic” trajectories xμ = x μ(s), with velocities uμ ≡ dx μ∕ds, characterized by vanishing accelerations

d2xν dxκ dxμ aν ≡ uμ∇ μu ν = ---2-− Γ νκμ --------= 0. (19 ) ds ds ds
A constant of motion 𝒳, such as those defined in Eq. (7), is conserved along a geodesic trajectory (19View Equation) in the sense that uμ∇ μ𝒳 = 0.

Circular geodesic motion in the equatorial plane (𝜃 = π∕2) is of fundamental importance in black hole accretion disk theory. The four velocity corresponding to circular motion is defined by,

μ μ μ u = A (η + Ωξ ), (20 )
where Ω = u ϕ∕ut = dϕ∕dt is the angular velocity measured by the stationary observer (ZAVO, see Section 2.2), and the redshift factor, A = ut, follows from uμu νg = − 1 μν,
− A −2 = gtt + 2Ωgt ϕ + Ω2g ϕϕ. (21 )
Other connections between these quantities that are particularly useful in our later calculations also follow from uμuνgμν = − 1:
ℓgtt + gtϕ Ωgϕϕ + gtϕ Ω = − ----------, ℓ = − ----------, ut = A (gtt + Ωgtϕ). (22 ) ℓgtϕ + gϕϕ Ωgtϕ + gtt
It is convenient to define the effective potential,
1- tt tϕ 2 ϕϕ 𝒰eff = − 2 ln |g − 2ℓg + ℓg |, (23 )
because in terms of 𝒰eff and the rescaled energy ∗ ℰ = ln ℰ, slightly non-circular motion, i.e., with V 2 = ururgrr + u𝜃u𝜃g𝜃𝜃 ≪ u ϕuϕgϕϕ, is characterized by the equation,
1- 2 ∗ 2V = ℰ − 𝒰eff, (24 )
which has the same form and the same physical meaning as the corresponding Newtonian equation. Therefore, exactly as in Newtonian theory, unperturbed circular Keplerian orbits are given by the condition of an extremum (minimum or maximum) of the effective potential (𝜃 = π∕2 ),
{( ) } { ( tt) ( tϕ) ( ϕϕ) } ∂-𝒰eff = 0 ⇒ ∂g-- − 2ℓ ∂g--- + ℓ2 ∂g--- = 0 . (25 ) ∂r ℓ ∂r ∂r ∂r
This quadratic equation for ℓ has two roots ℓ = ± ℓK(r,a), corresponding to “corotating” and “counterrotating” Keplerian orbits. Their explicit algebraic form is given in Eq. (35View Equation) in Section 2.5.

As in Newtonian theory, slightly non-circular orbits (with V ⁄= 0 being either δ˙r or δ ˙𝜃) are fully determined by the simple harmonic oscillator equations,

2 ¨ 2 δ¨r + ω rδr = 0, δ𝜃 + ω𝜃δ 𝜃 = 0, (26 )
where the radial ωr and vertical ω 𝜃 epicyclic frequencies are second derivatives of the effective potential,
( ) ( ) 2 ∂2-𝒰eff 2 ∂2-𝒰eff ωr = ∂r 2 , ω 𝜃 = ∂𝜃 2 , (27 ) ∗ ℓ ∗ ℓ
where ∂x∗2 = − gxx∂x2. The epicyclic frequencies (27View Equation) are measured by the comoving observer. To get the frequencies ω∗x measured by the stationary “observer at infinity” (Section 2.2), one must rescale by the redshift factor ω∗ = A ω x x. Obviously, when (ω )2 < 0 r, the epicyclic radial oscillations described by Eq. (26View Equation) are unstable – from Eq. (27View Equation) we see that they correspond to maxima of the effective potential. This happens for all circular orbits with radii less than r = rms (a ), and this limiting radius is called ISCO, the innermost stable circular orbit.
View Image

Figure 3: Evidence for the existence of the ISCO from data recorded by the Rossi X-ray Timing Explorer satellite from neutron star binary source 4U 1636–536 [33Jump To The Next Citation Point]. The source shows quasi-periodic oscillations (QPOs) with frequencies in the range 650 Hz < ν < 900 Hz. The sharp drop in the quality factor (bottom panel) seen at ∼ 870 Hz may be attributable to the ISCO [34Jump To The Next Citation Point].

Free circular orbits with r > rms are stable, while those with r < rms are not. Accordingly, accretion flows of almost free matter (i.e., with stresses insignificant in comparison with gravity or centrifugal effects), resemble almost circular motion for r > rms, and almost radial free-fall for r < rms. For thin disks, this transition in the character of the flow is expected to produce an effective inner truncation radius in the disk (see Section 5.3). The exceptional stability of the inner radius of the X-ray binary LMC X-3 [293], provides considerable evidence for such a connection and, hence, for the existence of the ISCO. The transition of the flow at the ISCO may also show up in the observed variability pattern, if variability is modulated by the orbital motion. In this case, one may expect that the there will be no variability observed with frequencies ν > νISCO, i.e., higher than the Keplerian orbital frequency at ISCO, or that the quality factor for variability, Q ∼ ν∕ Δν will significantly drop at νISCO. Several variants of this idea have been discussed [33Jump To The Next Citation Point, 34Jump To The Next Citation Point], and some observational evidence to support them has been presented (see Figure 3View Image).

2.4 The Paczyński–Wiita potential

For a non-rotating black hole (a = 0), the Kerr metric reduces to the Schwarzschild solution,

( ) ( )− 1 2 2M-- 2 2M-- 2 2[ 2 2 2] ds = − 1 − r dt + 1 − r dr + r d𝜃 + sin 𝜃d ϕ . (28 )
Paczyński and Wiita [236Jump To The Next Citation Point] proposed a practical and accurate Newtonian model for a Schwarzschild black hole, based on the gravitational potential,
GM 2GM ΦPW = − -------, rS = ---2--. (29 ) r − rS c
The Paczyński–Wiita potential became a very handy tool for studying black hole astrophysics. It has been used in many papers on the subject and still has applicability today. The Schwarzschild and the Paczyński–Wiita expressions for the Keplerian angular momentum and locations of the marginally stable and marginally bound orbits (Section 2.3) are identical. Similar, though less commonly adopted, pseudo-Newtonian potentials have also been found for Kerr (rotating) black holes [23, 276, 213].

2.5 Summary: characteristic radii and frequencies

We end this section with a few formulae for the Kerr geometry that we will use elsewhere in this review.

Keplerian circular orbits exist in the region r > r ph, with r ph being the circular photon orbit. Bound orbits exist in the region r > rmb, with rmb being the marginally bound orbit, and stable orbits exist for r > rms, with rms being the marginally stable orbit (also called the ISCO – Section 2.3). The location of these radii, as well as the location of the horizon rH and ergosphere r0, are given by the following formulae [31Jump To The Next Citation Point]:

{ [ ]} 2 −1 photon rph = 2rG 1 + cos 3-cos (a∗) , (30 ) ( √ ------) bound rmb = 2rG 1 − a∗-+ 1 − a∗ , (31 ) { 2 } stable r = r 3 + Z − [(3 − Z )(3 + Z + 2Z )]1∕2 , (32 ) ms G ( 2 ) 1 1 2 ∘ -----2 horizon rH = rG 1 + 1 − a∗ , (33 ) ( ∘ -----2---2--) ergosphere r0 = rG 1 + 1 − a∗cos 𝜃 , (34 )
where 2 1∕3 1∕3 1∕3 Z1 = 1 + (1 − a∗) [(1 + a∗) + (1 − a∗) ], 2 2 1∕2 Z2 = (3a∗ + Z1), a ∗ = a ∕M, and 2 rG = GM ∕c is the gravitational radius.

The Keplerian angular momentum ℓK and angular velocity ΩK, and the angular velocity of frame dragging ω are given by,

M 1∕2(r2 − 2aM 1∕2r1∕2 + a2) ℓK = ℓK (r,a) = ---3∕2-------1∕2------1∕2--, (35 ) r − 2M r + aM Ω2 = ------GM--------, (36 ) K (r3∕2 + aM 1∕2)2 (37 ) ω = ----------2aM--r----------. (38 ) (a2 + r2)ϱ2 + 2a2M r sin2 𝜃
The epicyclic frequencies measured “at infinity” are (here x = r∕M),
(ω∗)2 = Ω2 (1 − 6x −1 + 8a x−3∕2 − 3a2x−2) , (39 ) r K ( ∗ ) ∗ (ω∗𝜃)2 = Ω2K 1 − 4a ∗x −3∕2 + 3a2∗x −2 . (40 )
Comparing the Keplerian and epicyclic frequencies and the characteristic radii between the Schwarzschild metric and the Paczyński–Wiita potential (Section 2.4), we find for the Schwarzschild metric,
Ω2 = GM---= (ω ∗)2, (ω∗)2 = Ω2 (1 − 6x −1), K r3 𝜃 r K and xms = 6, xmb = 4, xph = 3, (41 )
and for the Paczyński–Wiita potential,
GM (1 − 6x−1) Ω2K = --------------= (ω∗𝜃)2, (ω ∗r)2 = Ω2K----------, r3(1 − 2x −1)2 (1 − 2x−1) and xms = 6, xmb = 4. (42 )

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