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Fundamental operation on complex numbers
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in
Complex_conjugate
Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m × n {\displaystyle m\times n} complex matrix A {\displaystyle
Conjugate_transpose
representation of it over the complex vector space V, then the complex conjugate representation Π is defined over the complex conjugate vector space V as follows:
Complex conjugate representation
Complex_conjugate_representation
Complex-differentiable (mathematical) function
{\displaystyle f} with respect to z ¯ {\displaystyle {\bar {z}}} , the complex conjugate of z {\displaystyle z} , is zero: ∂ f ∂ z ¯ = 0 , {\displaystyle
Holomorphic_function
Theorem about polynomials
In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P
Complex conjugate root theorem
Complex_conjugate_root_theorem
Mathematics concept
In mathematics, the complex conjugate of a complex vector space V {\displaystyle V\,} is a complex vector space V ¯ {\displaystyle {\overline {V}}} that
Complex conjugate of a vector space
Complex_conjugate_of_a_vector_space
Operation in complex geometry
In complex geometry, the complex conjugate line of a straight line is the line that it becomes by taking the complex conjugate of each point on this line
Complex_conjugate_line
Roots of an algebraic element's minimal polynomial
generalizes complex conjugation, since the algebraic conjugates over R {\displaystyle \mathbb {R} } of a complex number are the number itself and its complex conjugate
Conjugate element (field theory)
Conjugate_element_(field_theory)
Number with a real and an imaginary part
real if and only if it equals its own conjugate. The unary operation of taking the complex conjugate of a complex number cannot be expressed by applying
Complex_number
Matrix equal to its conjugate-transpose
self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose; that is, if the element in the j-th
Hermitian_matrix
Conjugate homogeneous additive map
: V → W {\displaystyle f:V\to W} between two complex vector spaces is said to be antilinear or conjugate-linear if f ( x + y ) = f ( x ) + f ( y ) (additivity)
Antilinear_map
"Bouncing back" of waves at an interface
the aberrating optics a second time. If one were to look into a complex conjugating mirror, it would be black because only the photons which left the
Reflection_(physics)
Mathematical operation in linear algebra
denotes the conjugate transpose of x {\displaystyle \mathbf {x} } (conjugate of the transpose, or equivalently transpose of the conjugate). Matrix multiplication
Matrix_multiplication
Vector space with generalized dot product
of F. A bar over an expression representing a scalar denotes the complex conjugate of this scalar. A zero vector is denoted 0 {\displaystyle \mathbf
Inner_product_space
Critical point where a periodic solution arises
system of differential equations. When this matrix has a pair of complex-conjugate eigenvalues that cross the imaginary axis as a parameter is varied
Hopf_bifurcation
Reals with an extra square root of +1 adjoined
1} . A split-complex number has two real number components x and y, and is written z = x + y j . {\displaystyle z=x+yj.} The conjugate of z is z ∗ =
Split-complex_number
Distance from zero to a number
{\displaystyle |z|=r.} Since the product of any complex number z {\displaystyle z} and its complex conjugate z ¯ = x − i y {\displaystyle {\bar {z}}=x-iy}
Absolute_value
Notation for quantum states
versa. The Hermitian conjugate of a complex number is its complex conjugate. The Hermitian conjugate of the Hermitian conjugate of anything (linear operators
Bra–ket_notation
Type of vector space in math
{x^{2}+y^{2}}}\,.} The inner product of a pair of complex numbers z and w is the product of z with the complex conjugate of w: ⟨ z , w ⟩ = z w ¯ . {\displaystyle
Hilbert_space
Complex exponential in terms of sine and cosine
when used in multiplication or powers of complex numbers. Any complex number z = x + iy, and its complex conjugate, z = x − iy, can be written as z = x +
Euler's_formula
Topics referred to by the same term
degree Conjugate transpose, the complex conjugate of the transpose of a matrix Harmonic conjugate in complex analysis Conjugate (graph theory), an alternative
Conjugation
Mathematical function that preserves angles
U {\displaystyle U} . If f {\displaystyle f} is antiholomorphic (complex conjugate to a holomorphic function), it preserves angles but reverses their
Conformal_map
Conjugate transpose of an operator in infinite dimensions
ft(A^{-1}\right)^{*}} Conjugate linearity: (A + B)∗ = A∗ + B∗ (λA)∗ = λA∗, where λ denotes the complex conjugate of the complex number λ "Anti-distributivity":
Hermitian_adjoint
Type of function in mathematics
analytic where the pieces meet. The complex conjugate function z → z ∗ {\displaystyle z\to z^{*}} is not complex analytic, although its restriction
Analytic_function
Polynomial equation of degree 3
non-real complex conjugate roots. This can be proved as follows. First, if r is a root of a polynomial with real coefficients, then its complex conjugate is
Cubic_equation
Number with an integer power equal to 1
exponents. In particular, the reciprocal of an nth root of unity is its complex conjugate, and is also an nth root of unity: 1 z = z − 1 = z n − 1 = z ¯ . {\displaystyle
Root_of_unity
Algebraic operation on coordinate vectors
{b_{i}}}},} where b i ¯ {\displaystyle {\overline {b_{i}}}} is the complex conjugate of b i {\displaystyle b_{i}} . When vectors are represented by column
Dot_product
Theorem in electrical engineering
maximum power transfer occurs when the load impedance is equal to the complex conjugate of the source impedance. The mathematics of the theorem also applies
Maximum power transfer theorem
Maximum_power_transfer_theorem
Characteristic property of holomorphic functions
suggests another interpretation of the Cauchy–Riemann equations. The complex conjugate of z {\displaystyle z} , denoted z ¯ {\textstyle {\bar {z}}} , is
Cauchy–Riemann_equations
Covariance and correlation
dt} where f ( t ) ¯ {\displaystyle {\overline {f(t)}}} denotes the complex conjugate of f ( t ) {\displaystyle f(t)} , and τ {\displaystyle \tau } is called
Cross-correlation
Concepts from linear algebra
The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs. The spectrum of a matrix is the
Eigenvalues_and_eigenvectors
Inequality between integrals in Lp spaces
measurable real- or complex-valued functions defined on S. If ‖fg‖1 is finite, then the pointwise products of f with g and its complex conjugate function are
Hölder's_inequality
Linear transformation of spacetime coordinates
be a 2 × 2 complex matrix with determinant 1 and let A † {\displaystyle A^{\dagger }} be the hermitian conjugate of A (the complex conjugate of the transpose
Biquaternion Lorentz transformation
Biquaternion_Lorentz_transformation
Branch of mathematics studying functions of a complex variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions
Complex_analysis
Method of drawing geometric objects
and closed under the complex conjugate and square root operations (to avoid ambiguity, we can specify the square root with complex argument less than π)
Straightedge and compass construction
Straightedge_and_compass_construction
mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , its conjugate g ¯ {\displaystyle
Complex_Lie_algebra
Polynomial equation of degree two
double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included
Quadratic_equation
Complex matrix whose conjugate transpose equals its inverse
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if U ∗ U = U
Unitary_matrix
Relativistic wave description of fermions
relates a four-component spinor to its charge conjugate. As a 2×2 differential equation acting on a complex two-component spinor, resembling the Weyl equation
Majorana_equation
Adjusting input/output impedances of an electrical circuit for some purpose
Z_{\text{load}}=Z_{\text{source}}^{*},} where a superscript * indicates the complex conjugate. A conjugate match is different from a reflection-less match when either
Impedance_matching
Property determining comparison and ordering
Alternatively, the magnitude of a complex number z may be defined as the square root of the product of itself and its complex conjugate, z ¯ {\displaystyle {\bar
Magnitude_(mathematics)
Function of the coefficients of a polynomial that gives information on its roots
has two distinct real roots, and negative if it has two distinct complex conjugate roots. Similarly, the discriminant of a cubic polynomial is zero if
Discriminant
derived from the two different conventions. The two definitions are complex conjugates of each other. One way to incorporate attenuation into the mathematical
Mathematical descriptions of opacity
Mathematical_descriptions_of_opacity
Horizontal line immediately above a portion of writing
-2+0.07918={\bar {2}}.07918} The overline notation can indicate a complex conjugate and analogous operations. if x = a + i b {\displaystyle x=a+ib} ,
Overline
Measure of the electric polarizability of a dielectric material
'-i\varepsilon ''} is the complex permittivity Note that this is using the electrical engineering convention of the complex conjugate ambiguity; the physics/chemistry
Permittivity
Branch of physics
{1}{2}}E_{2}e^{-i\omega _{2}t}+{\text{c.c.}},} where "c.c." stands for complex conjugate. Plugging this into the expression for P gives P NL = ε 0 χ ( 2 )
Nonlinear_optics
Number which when multiplied by x equals 1
z ¯ ‖ z ‖ {\displaystyle {\tfrac {\bar {z}}{\|z\|}}} gives us the complex conjugate with a magnitude reduced to a value of 1, so dividing again by |z|
Multiplicative_inverse
Horizontal line used in mathematical notation
1428571428571428571... a + b i ¯ {\displaystyle {\overline {a+bi}}} complex conjugate Y = A B ¯ {\displaystyle Y={\overline {AB}}} boolean NOT (A AND B)
Vinculum_(symbol)
Physical characteristic of oscillating systems
pole on the complex plane and the damping ratio of that pole determines how quickly that oscillation decays. In general, Complex conjugate pairs of poles
Resonance
Curve from a cone intersecting a plane
conic section are real, the points at infinity are either real or complex conjugate. What should be considered as a degenerate case of a conic depends
Conic_section
Polynomial function of degree 4
follows: If ∆ < 0 then the equation has two distinct real roots and two complex conjugate non-real roots. If ∆ > 0 then either the equation's four roots are
Quartic_function
Matrix operation which flips a matrix over its diagonal
^{\text{T}}=-\mathbf {A} .} A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline)
Transpose
Mathematical function, denoted exp(x) or e^x
{\displaystyle e^{i\pi }=-1} and the functional identity. The complex conjugate of the complex exponential is e z ¯ = e z ¯ . {\displaystyle {\overline
Exponential_function
Non-tensorial representation of the spin group
+ i v {\displaystyle z=u+iv} is complex, and z ¯ = u − i v {\displaystyle {\bar {z}}=u-iv} is the complex conjugate of z {\displaystyle z} . Then H 2
Spinor
Every polynomial has a real or complex root
product of a polynomial and its complex conjugate (obtained by replacing each coefficient with its complex conjugate). A root of this product is either
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Polynomial equation of degree 4
real solutions – then there is another complex solution x {\displaystyle x} 2 which is the complex conjugate of x {\displaystyle x} 1. If the other two
Quartic_equation
Polynomial with reversed root positions
complex numbers, when p ( z ) = a 0 + a 1 z + a 2 z 2 + ⋯ + a n z n , {\displaystyle p(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots +a_{n}z^{n},} the conjugate reciprocal
Reciprocal_polynomial
the Hermitian conjugate (also called the conjugate transpose) of A {\displaystyle A} , defined as applying both the complex conjugate and the transpose
+_h.c.
Number whose cube is a given number
real, one of the cube roots is real and the two other are nonreal complex conjugate numbers. Otherwise, the three cube roots are all nonreal. For example
Cube_root
Symmetry of physical laws under a charge-conjugation transformation
ultimately revealed to be a symmetry under complex conjugation, although exactly what is being conjugated where can be at times obfuscated, depending
C-symmetry
Formula that provides the solutions to a quadratic equation
the equation has no real roots but has two distinct complex roots, which are complex conjugates of each other. Geometrically, the roots represent the
Quadratic_formula
{\overline {\Box }}} 1. Complex conjugate: If z is a complex number, then z ¯ {\displaystyle {\overline {z}}} is its complex conjugate. For example, a + b
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Geometric representation of the complex numbers
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal x-axis, called
Complex_plane
2nd-degree plane curve which is reducible
single point (in fact, two complex conjugate lines), or the null set (twice the line at infinity or two parallel complex conjugate lines). All these degenerate
Degenerate_conic
Values which describe behavior of a linear electric circuit
is the complex conjugate of Z i {\displaystyle Z_{i}} , V i {\displaystyle V_{i}} and I i {\displaystyle I_{i}} are respectively the complex amplitudes
Scattering_parameters
Type of representation in representation theory
act either on real or complex column vectors. A real representation on a complex vector space is isomorphic to its complex conjugate representation, but
Real_representation
Matrix decomposition method
positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte
Cholesky_decomposition
Length in a vector space
denotes its conjugate transpose. This formula is valid for any inner product space, including Euclidean and complex spaces. For complex spaces, the inner
Norm_(mathematics)
Quadric surface with one axis of symmetry and no center of symmetry
hyperbolic if the factors are real; elliptic if the factors are complex conjugate. An elliptic paraboloid is shaped like an oval cup and has a maximum or
Paraboloid
Filters used in signal processing that are optimal in some sense
unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template. The matched filter is the optimal
Matched_filter
Type of complex function
mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable
Hermitian_function
Mathematical relation defining a sequence
instead there may be some that are complex numbers. In the latter case, all the complex roots come in complex conjugate pairs. If all the characteristic
Linear recurrence with constant coefficients
Linear_recurrence_with_constant_coefficients
Relativistic wave equation describing massless fermions
transformation. The skew complex conjugate ω ψ ∗ = i σ 2 ψ {\displaystyle \omega \psi ^{*}=i\sigma ^{2}\psi } can be recognized as the charge conjugate form of ψ
Weyl_equation
Generalization of complex inner products
{\overline {a}}} is the complex conjugate of a scalar a . {\displaystyle a.} A complex sesquilinear form can also be viewed as a complex bilinear map V ¯ ×
Sesquilinear_form
Concept in complex analysis
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. It is the simplest
Zeros_and_poles
Irreducible representation of the rotation group SO
fundamental role in the quantum mechanical theory of angular momentum. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical
Wigner_D-matrix
Fourier transform of the probability density function
{t}}} is the complex conjugate of t {\textstyle t} and Re ( z ) {\textstyle \operatorname {Re} (z)} is the real part of the complex number z {\textstyle
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Probability distribution on complex matrices
column p-vector of random complex Gaussian zero-mean samples and ( . ) H {\displaystyle (.)^{H}} is an Hermitian (complex conjugate) transpose. If the covariance
Complex_Wishart_distribution
Mathematics concept
applications of these ideas. Almost complex manifold Complex manifold Complex differential form Complex conjugate vector space Hermitian structure Real
Linear_complex_structure
Topics referred to by the same term
the combustor Complex conductivity (measurement method), a measurement method in geophysics Complex conjugate, an operation on complex numbers, commonly
CC
Method for producing composition algebras
extensions of their complex analogs: if a and b are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has
Cayley–Dickson_construction
Change of the sign of a square root
In mathematics, the conjugate of an expression of the form a + b d {\displaystyle a+b{\sqrt {d}}} is a − b d , {\displaystyle a-b{\sqrt {d}},} provided
Conjugate_(square_roots)
Attribute of a mathematical function
In mathematics, more specifically complex analysis, the residue of a function at a point of its domain is a complex number proportional to the contour
Residue_(complex_analysis)
Sum in algebraic number theory
character χ the equation relating L(s, χ) and L(1 − s, χ) (where χ is the complex conjugate of χ) involves a factor[clarification needed] G ( χ ) | G ( χ ) |
Gauss_sum
other words, the group elements are expressed as complex matrices, and the complex conjugate of a complex representation is a different, non-equivalent representation
Complex_representation
Equation describing the transport of some quantity
Schrödinger equation The time dependent Schrödinger equation and its complex conjugate (i → −i throughout) are respectively: − ℏ 2 2 m ∇ 2 Ψ + U Ψ = i ℏ
Continuity_equation
Matrix that commutes with its conjugate transpose
In mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A*: A normal ⟺ A ∗ A = A A ∗ . {\displaystyle A{\text{
Normal_matrix
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
\lambda } cancels with its complex conjugate λ ∗ {\displaystyle \lambda ^{*}} in both parts of p {\displaystyle p} : in the complex 2 z 0 z 1 ∗ {\displaystyle
Hopf_fibration
Measure used in radio engineering and telecommunications
Impedance matching is achieved when the source impedance is the complex conjugate of the load impedance. The easiest way of achieving this, and the
Standing_wave_ratio
Representation of a type of random process
autocorrelation function that decays exponentially. Similarly, each pair of complex conjugate roots contributes an exponentially damped oscillation. The simplest
Autoregressive_model
In mathematics, invariant of square matrices
determinant of the complex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant
Determinant
Dual pair of vector spaces
{\displaystyle H.} Let H ¯ {\displaystyle {\overline {H}}} denote the complex conjugate vector space of H , {\displaystyle H,} where H ¯ {\displaystyle {\overline
Dual_system
Typographical symbol (*)
a superscript The complex conjugate of a complex number (the more common notation is z ¯ {\displaystyle {\bar {z}}} ). The conjugate in a composition algebra
Asterisk
Property of a differential manifold that includes complex structures
with its complex conjugate is the zero section: L ∩ L ¯ = 0 {\displaystyle L\cap {\overline {L}}=0} ; L is maximal isotropic, i.e. its complex rank equals
Generalized_complex_structure
Function that is holomorphic on the whole complex plane
arguments, and the value of the function at the complex conjugate of z {\displaystyle z} will be the complex conjugate of the value at z {\displaystyle z}
Entire_function
(Mathematical) decomposition into a product
factors, one has to replace every pair of complex conjugate factors by its product. As the complex conjugate of e i α {\displaystyle e^{i\alpha }} is e
Factorization
Array of numbers
conjugate transpose of the matrix, that is, the transpose of the complex conjugate of A. By the spectral theorem, real symmetric matrices and complex
Matrix_(mathematics)
Provides integral formulas for all derivatives of a holomorphic function
formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on
Cauchy's_integral_formula
Number, approximately 1.618
quadratic integers, however the sum of any fifth root of unity and its complex conjugate, z + z ¯ {\displaystyle z+{\bar {z}}} , is a quadratic integer
Golden_ratio
Measure of covariance of components of a random vector
where the complex conjugate of a complex number z {\displaystyle z} is denoted z ¯ {\displaystyle {\overline {z}}} ; thus the variance of a complex random
Covariance_matrix
COMPLEX CONJUGATE
COMPLEX CONJUGATE
Girl/Female
Bengali, Indian
Good Complex
Girl/Female
Tamil
Complete
Boy/Male
Tamil
Poornan | பூரà¯à®¨à®¾à®¨
Complete
Poornan | பூரà¯à®¨à®¾à®¨
Boy/Male
Indian
Complete
Boy/Male
Tamil
Complete
Surname or Lastname
English (Yorkshire)
English (Yorkshire) : habitational name from any of various places called Copley, for example in County Durham, Staffordshire, and Yorkshire, from the Old English personal name Coppa (apparently a byname for a tall man) or from copp ‘hilltop’ + lēah ‘woodland clearing’.
Surname or Lastname
English
English : unexplained.Americanized form of German Koppler.
Boy/Male
Indian
Complete
Girl/Female
Tamil
Shesha Harani | ஷேஷ ஹரணீÂ
Complete
Shesha Harani | ஷேஷ ஹரணீÂ
Boy/Male
Tamil
Complete
Girl/Female
Tamil
Complete
Surname or Lastname
English
English : habitational name, probably from Comley in Shropshire or Combley on the Isle of Wight; both are named with Old English cumb ‘valley’ + lēah ‘woodland clearing’.
Girl/Female
Tamil
Complete
Girl/Female
Hindu, Indian
Complex
Girl/Female
Tamil
Complete
Girl/Female
Tamil
Sompurna | ஸோமபà¯à®°à¯à®¨à®¾
Complete
Sompurna | ஸோமபà¯à®°à¯à®¨à®¾
Girl/Female
Arabic, Muslim
Complex; Zigzag; Curling
Surname or Lastname
English
English : habitational name from Coppull in Lancashire, recorded in the 13th century as Cophill, from Old English copp ‘peak’ + hyll ‘hill’.English : nickname from Old French curt peil ‘short hair’.Probably an Americanized spelling of German and Jewish Koppel or German and Dutch Kappel.
Girl/Female
Muslim
Complex, Zigzag, Curling
Boy/Male
Tamil
Complete
COMPLEX CONJUGATE
COMPLEX CONJUGATE
Girl/Female
American, Christian, Dutch, French, German, Greek, Hungarian, Irish, Latin, Netherlands
Pearl
Male
Esperanto
Modern Esperanto form of Hebrew Miyka'el, MIKELO means "who is like God?"
Girl/Female
Hindu
World
Surname or Lastname
English
English : patronymic from Jenkin. Jenkins is one of the most common names in England, especially southwestern England, but is also especially associated with Wales.
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
Iris
Girl/Female
Hindu, Indian, Marathi
Goddess of Vegetation
Boy/Male
Indian, Tamil
Honest Person
Girl/Female
Hindu, Indian, Marathi
Crest Jewel; Disciplined; Cultured
Girl/Female
Hindu, Indian, Sindhi
Guide to Righteousness
Girl/Female
Tamil
Praharshini | பà¯à®°à®¹à®¾à®°à¯à®·à¯€à®¨à¯€
One who makes others Happy
COMPLEX CONJUGATE
COMPLEX CONJUGATE
COMPLEX CONJUGATE
COMPLEX CONJUGATE
COMPLEX CONJUGATE
a.
Complex, complicated.
imp. & p. p.
of Comply
a.
See Couple-close.
a.
One of the pairs of plates of two metals which compose a voltaic battery; -- called a voltaic couple or galvanic couple.
a.
Not complex; uncompounded; simple.
n.
A complex; an aggregate of parts; a complication.
n.
Composed of two or more parts; composite; not simple; as, a complex being; a complex idea.
a.
Finished; ended; concluded; completed; as, the edifice is complete.
pl.
of Couple-close
n.
One who couples; that which couples, as a link, ring, or shackle, to connect cars.
imp. & p. p.
of Compile
a.
Repeatedly compound; made up of complex constituents.
adv.
In a complex manner; not simply.
a.
Intricate; entangled; complicated; complex.
a.
That which joins or links two things together; a bond or tie; a coupler.
imp. & p. p.
of Couple
n.
Two taken together; a pair or couple; especially two lines of verse that rhyme with each other.
v. t.
To bring to a state in which there is no deficiency; to perfect; to consummate; to accomplish; to fulfill; to finish; as, to complete a task, or a poem; to complete a course of education.
n.
One who compiles; esp., one who makes books by compilation.
n.
One who complies, yields, or obeys; one of an easy, yielding temper.