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Figure 1. Lucifer (the morning star). Engraving by G.H. Frezza, 1704

Lucifer (/ˈluːsɪfər/;[1][2][3] LOO-sif-ər) is the King James Version rendering of the Hebrew word הֵילֵל in Isaiah (Isaiah 14:12). The Vulgate translation uses the Latin word lucifer, but with a lower-case initial.[4] The Hebrew word, transliterated Hêlêl[5] or Heylel (pron. as HAY-lale),[6] occurs once in the Hebrew Bible[5] and according to the KJV-based Strong's Concordance means "shining one, light-bearer".[6] The Septuagint renders הֵילֵל in Greek as ἑωσφόρος[7][8][9][10][11] (heōsphoros), a name, literally "bringer of dawn", for the morning star. The word Lucifer is taken from the Latin Vulgate,[16] which translates הֵילֵל as lucifer,[17][18] meaning "the morning star, the planet Venus", or, as an adjective, "light-bringing".[19]


Later Christian tradition came to use the Latin word for "morning star", lucifer, as a proper name ("Lucifer") for the devil; as he was before his fall. As a result, "'Lucifer' has become a by-word for Satan / the Devil in the church and in popular literature", as in Dante Alighieri's Inferno, Joost van den Vondel's Lucifer and John Milton's Paradise Lost. However, the Latin word never came to be used almost exclusively, as in English, in this way, and was applied to others also, including Jesus. The image of a morning star fallen from the sky is generally believed among scholars to have a parallel in Canaanite mythology.[22]
However, according to both Christian and Jewish exegesis, in the Book of Isaiah, chapter 14, the King of Babylon, Nebuchadnezzar II, conqueror of Jerusalem, is condemned in a prophetic vision by the prophet Isaiah and is called the "Morning Star" (planet Venus). In this chapter the Hebrew text says הֵילֵל בֶּן-שָׁחַר (Helel ben Shachar, "shining one, son of the morning"). "Helel ben Shahar" may refer to the Morning Star, but the text in Isaiah 14 gives no indication that Helel was a star or planet.[27][28]

1 Etymology
2 Mythology
3 Literal meaning
4 Latin
5 Apocrypha and pseudepigrapha
6 Christianity
6.1 Lucifer as Satan or the devil
7 Islam
8 Anthroposophy
9 Occultism
10 Taxil's hoax

Illustration of Lucifer in the first fully illustrated print edition of Dante Alighieri's Divine Comedy. Woodcut for Inferno, canto 33. Pietro di Piasi, Venice, 1491. The translation of הֵילֵל as "Lucifer", as in the King James Version, has been abandoned in modern English translations of Isaiah 14:12. Present-day translations have "morning star" (New International Version, New Century Version, New American Standard Bible, Good News Translation, Holman Christian Standard Bible, Contemporary English Version, Common English Bible, Complete Jewish Bible), "daystar" (New Jerusalem Bible, English Standard Version, The Message, "Day Star" New Revised Standard Version), "shining one" (New Life Version, New World Translation, JPS Tanakh) or "shining star" (New Living Translation).

The term appears in the context of an oracle against a dead king of Babylon,[29] who is addressed as הילל בן שחר (Hêlêl ben Šāḥar),[30][31] rendered by the King James Version as "O Lucifer, son of the morning!" and by others as "morning star, son of the dawn".

In a modern translation from the original Hebrew, the passage in which the phrase "Lucifer" or "morning star" occurs begins with the statement: "On the day the Lord gives you relief from your suffering and turmoil and from the harsh labour forced on you, you will take up this taunt against the king of Babylon: How the oppressor has come to an end! How his fury has ended!"[32] After describing the death of the king, the taunt continues:

"How you have fallen from heaven, morning star, son of the dawn! You have been cast down to the earth, you who once laid low the nations! You said in your heart, 'I will ascend to the heavens; I will raise my throne above the stars of God; I will sit enthroned on the mount of assembly, on the utmost heights of Mount Zaphon. I will ascend above the tops of the clouds; I will make myself like the Most High.' But you are brought down to the realm of the dead, to the depths of the pit. Those who see you stare at you, they ponder your fate: 'Is this the man who shook the earth and made kingdoms tremble, the man who made the world a wilderness, who overthrew its cities and would not let his captives go home?'"[33] J. Carl Laney has pointed out that in the final verses here quoted, the king of Babylon is described not as a god or an angel but as a man; and that man may have been not Nebuchadnezzar II, but rather his son, Belshazzar. During the trito Isaiah period of the Persian sacking of the Neo-Babylonian Empire, Nebuchadnezzar was gripped by a spiritual fervor to build a temple to the moon god Sin (possibly analogous with Hubal, the primary God of pre-Islamic Mecca), and his son ruled as regent. The Abrahamic scriptural texts could be interpreted as a weak usurping of true kingly power, and a taunt at the failed regency of Belshazzar.[34][35]

For the unnamed "king of Babylon" a wide range of identifications have been proposed. They include a Babylonian ruler of the prophet Isaiah's own time[37] the later Nebuchadnezzar II, under whom the Babylonian captivity of the Jews began, or Nabonidus,[37][38] and the Assyrian kings Tiglath-Pileser, Sargon II and Sennacherib.[34][37][39] Herbert Wolf held that the "king of Babylon" was not a specific ruler but a generic representation of the whole line of rulers.[40]

In ancient Canaanite mythology, the morning star is pictured as a god, Attar, who attempted to occupy the throne of Ba'al and, finding he was unable to do so, descended and ruled the underworld.[41][42] The original myth may have been about a lesser god Helel trying to dethrone the Canaanite high god El who lived on a mountain to the north.[43][44] Hermann Gunkel's reconstruction of the myth told of a mighty warrior called Hêlal, whose ambition it was to ascend higher than all the other stellar divinities, but who had to descend to the depths; it thus portrayed as a battle the process by which the bright morning star fails to reach the highest point in the sky before being faded out by the rising sun.[45]


Planet Venus rising above the horizon at dawn Similarities have been noted with the East Semitic story of Ishtar's or Inanna's descent into the underworld, Ishtar and Inanna being associated with the planet Venus.[46] A connection has been seen also with the Babylonian myth of Etana. The Jewish Encyclopedia comments:

"The brilliancy of the morning star, which eclipses all other stars, but is not seen during the night, may easily have given rise to a myth such as was told of Ethana and Zu: he was led by his pride to strive for the highest seat among the star-gods on the northern mountain of the gods ... but was hurled down by the supreme ruler of the Babylonian Olympus."[47] The Greek myth of Phaethon, whose name, like that of הֵילֵל, means "Shining One", has also been seen as similar.

The Eerdmans Commentary on the Bible points out that no evidence has been found of any Canaanite myth of a god being thrown from heaven, as in Isaiah 14:12. It concludes that the closest parallels with Isaiah's description of the king of Babylon as a fallen morning star cast down from heaven are to be found not in any lost Canaanite and other myths but in traditional ideas of the Jewish people themselves, echoed in the Biblical account of the fall of Adam and Eve, cast out of God's presence for wishing to be as God, and the picture in Psalm 82 of the "gods" and "sons of the Most High" destined to die and fall. This Jewish tradition has echoes also in Jewish pseudepigrapha such as 2 Enoch and the Life of Adam and Eve.[29][47][48]

Literal meaning
See also: Satan in Judaism
The Hebrew words הֵילֵל בֶּן-שָׁחַר (Helel ben Shaḥar, "day-star, son of the morning") in Isaiah 14:12 are part of a prophetic vision against an oppressive king of Babylon.[49] Jewish exegesis of Isaiah 14:12–15 identified the king of Babylon as Nebuchadnezzar II.[50] Verse 20 says that this king of Babylon will not be "joined with them [all the kings of the nations] in burial, because thou hast destroyed thy land, thou hast slain thy people; the seed of evil-doers shall not be named for ever", but rather be cast out of the grave, while "All the kings of the nations, all of them, sleep in glory, every one in his own house".[28][51]

As an adjective, the Latin word lucifer meant "light-bringing" and was applied to the moon. As a noun, it meant "morning star", or, in Roman mythology, its divine personification as "the fabled son of Aurora[52] and Cephalus, and father of Ceyx", or (in poetry) "day".[19] The second of the meanings attached to the word when used as a noun corresponds to the image in Greek mythology of Eos, the goddess of dawn, giving birth to the morning star Phosphorus.[52]

Isaiah 14:12 is not the only place where the Vulgate uses the word lucifer. It uses the same word four more times, in contexts where it clearly has no reference to a fallen angel: 2 Peter 1:19 (meaning "morning star"), Job 11:17 ("the light of the morning"), Job 38:32 ("the signs of the zodiac") and Psalms 110:3 ("the dawn").[53] Lucifer is not the only expression that the Vulgate uses to speak of the morning star: three times it uses stella matutina: Sirach 50:6 (referring to the actual morning star), and Revelation 2:28 (of uncertain reference) and 22:16 (referring to Jesus).

Indications that in Christian tradition the Latin word lucifer, unlike the English word, did not necessarily call a fallen angel to mind exist also outside the text of the Vulgate. Two bishops bore that name: Saint Lucifer of Cagliari, and Lucifer of Siena.

In Latin, the word is applied to John the Baptist and is used as a title of Jesus himself in several early Christian hymns. The morning hymn Lucis largitor splendide of Hilary contains the line: "Tu verus mundi lucifer" (you are the true light bringer of the world).[54] Some interpreted the mention of the morning star (lucifer) in Ambrose's hymn Aeterne rerum conditor as referring allegorically to Jesus and the mention of the cock, the herald of the day (praeco) in the same hymn as referring to John the Baptist.[55] Likewise, in the medieval hymn Christe qui lux es et dies, some manuscripts have the line "Lucifer lucem proferens".[56]

The Latin word lucifer is also used of Jesus in the Easter Proclamation prayer to God regarding the paschal candle: Flammas eius lucifer matutinus inveniat: ille, inquam, lucifer, qui nescit occasum. Christus Filius tuus, qui, regressus ab inferis, humano generi serenus illuxit, et vivit et regnat in saecula saeculorum ("May this flame be found still burning by the Morning Star: the one Morning Star who never sets, Christ your Son, who, coming back from death's domain, has shed his peaceful light on humanity, and lives and reigns for ever and ever"). In the works of Latin grammarians, Lucifer, like Daniel, was discussed as an example of a personal name.[57]

Apocrypha and pseudepigrapha
In the Second Temple period literature the main possible reference is found in 2 Enoch, also known as Slavonic Enoch:

2 Enoch 29:3 Here Satanail was hurled from the height together with his angels

However the editor of the standard modern edition (Charlesworth, Old Testament Pseudepigrapha Vol.1) pipelines the verse as a probable later Christian interpolation on the grounds that "Christian explanations of the origin of evil linked Lk 10:18 with Isa 14 and eventually Gen. 3 so vs 4 could be a Christian interpolation... Jewish theology concentrated on Gen 6., and this is prominent in the Enoch cycle as in other apocalypses." Furthermore, the name used in 2 Enoch, Satanail, is not directly related to the Isaiah 14 text, and the surrounding imagery of fire suggests Ezekiel 28:17–18.

Other instances of lucifer in the Old Testament pseudepigrapha are related to the "star" Venus, in the Sibylline Oracles battle of the constellations (line 517) "Lucifer fought mounted on the back of Leo",[59] or the entirely rewritten Christian version of the Greek Apocalypse of Ezra 4:32 which has a reference to Lucifer as Antichrist.[60]

An association of Isaiah 14:12–18 with a personification of evil, called the devil developed outside of mainstream Rabbinic Judaism in pseudepigrapha and Christian writings,[61] particularly with the apocalypses.[62]

Especially Isaiah 14:12, became a dominant conception of a fallen angel motif in 1 Enoch 86-90 and 2 Enoch 29:3–4. Rabbinical Judaism rejected any belief in rebel or fallen angels.[64] In the 11th century, the Pirqe de-Rabbi Eliezer illustrates the origin of the "fallen angel myth" by giving two accounts, one relates to the angel in the Garden of Eden who seduces Eve, and the other relates to the angels, the benei elohim who cohabit with the daughters of man (Genesis 6:1–4).[65]

Main article: Devil in Christianity
Christian writers applied the words of Isaiah 14:12 to Satan. Sigve K Tonstad argues that the New Testament War in Heaven theme of Revelation 12:7-9, in which the dragon "who is called the devil and Satan … was thrown down to the earth", derives from the passage in Isaiah 14.[66] Origen (184/185 – 253/254) interpreted such Old Testament passages as being about manifestations of the Devil; but of course, writing in Greek, not Latin, he did not identify the devil with the name "Lucifer".[67][68][69][70] Tertullian (c. 160 – c. 225), who wrote in Latin, also understood Isaiah 14:14 ("I will ascend above the tops of the clouds; I will make myself like the Most High") as spoken by the Devil,[71] but "Lucifer" is not among the numerous names and phrases he used to describe the devil.[72] Even at the time of the Latin writer Augustine of Hippo (354–430), "Lucifer" had not yet become a common name for the Devil.[67]

Some time later, the metaphor of the morning star that Isaiah 14:12 applied to a king of Babylon gave rise to the general use of the Latin word for "morning star", capitalized, as the original name of the devil before his fall from grace, linking Isaiah 14:12 with Luke 10:18 ("I saw Satan fall like lightning from heaven") and interpreting the passage in Isaiah as an allegory of Satan's fall from heaven.[73][74]

However, the understanding of the morning star in Isaiah 14:12 as a metaphor referring to a king of Babylon continued also to exist among Christians. Theodoret of Cyrus (c. 393 – c. 457) wrote that Isaiah calls the king "morning star", not as being the star, but as having had the illusion of being it.[75] The same understanding is shown in Christian translations of the passage, which in English generally use "morning star" rather than treating the word as a proper name, "Lucifer". So too in other languages, such as French,[76] German,[77] Portuguese, and Spanish. Even the Vulgate text in Latin is printed with lower-case lucifer (morning star), not upper-case Lucifer (proper name).[18]

Calvin said: "The exposition of this passage, which some have given, as if it referred to Satan, has arisen from ignorance: for the context plainly shows these statements must be understood in reference to the king of the Babylonians."[80] Luther also considered it a gross error to refer this verse to the devil.[81]

Gustave Doré, illustration to Paradise Lost, book IX, 179–187: "... he [Satan] held on /His midnight search, where soonest he might finde /The Serpent: him fast sleeping soon he found ..." Lucifer as Satan or the devil
Adherents of the King James Only movement and others who hold that Isaiah 14:12 does indeed refer to the devil have decried the modern translations.

Treating "Lucifer" as a name for the devil or Satan, they may use that name when speaking of such accounts of the devil or Satan as the following:

Satan inciting David to number Israel (1 Chronicles 21:1), though in 2 Samuel 24:1 it is stated that God caused David to take census of Israel, possibly pointing to a deeply rooted Gnostic belief in which the archons ascribed to Satan and Jehovah are merely archons—a dualist expression of the Monad's will, and part of the demiurge. Job tested by Satan (Book of Job) Satan ready to accuse the high priest Joshua (Zechariah 3:1–2) Sin brought into the world through the devil's envy (Wisdom 2:24) "The prince of the power of the air, the spirit that is now at work in the sons of disobedience" (Ephesians 2:2) "The god of this world" (2 Corinthians 4:4). The devil disputing with Michael about the body of Moses (Jude 1:9) The dragon of the Book of Revelation "who is called the devil and Satan" (Revelation 12:9;20:2) They may also use the name Lucifer when speaking of Satan's motive for rebelling and of the nature of his sin, which, without using the name Lucifer, Origen, Chrysostom, Jerome, Ambrose, and Augustine attributed to the devil's pride, and Irenaeus, Tertullian, Justin Martyr, Cyprian, and again Augustine attributed to the devil's envy of humanity created in the image of God.[85][86][87] Jealousy of humans, created in the divine image and given authority over the world is the motive that a modern writer, who denies that there is any such person as Lucifer, says that Tertullian attributed to the devil, and, while he cited Tertullian and Augustine as giving envy as the motive for the fall, an 18th-century French Capuchin preacher himself described the rebel angel as jealous of Adam's exaltation, which he saw as a diminution of his own status.[87]

Main article: Devil (Islam)
In Islam the Devil is known as Iblīs (Arabic: إبليس‎‎, plural: ابالسة abālisah) or Shaytān (Arabic: شيطان‎‎, plural: شياطين shayātīn). Iblis is banished from heaven for refusing to prostrate himself before Adam, which is similar to the earlier 3 Enoch, chapter 4, in which all of the angels prostrate themselves before Enoch, an early descendant of Adam. Thus, he sins after the creation of man. He asks God for a respite until judgment day rather than being consigned to the fire of hell immediately. God grants this request, and Iblis then swears revenge by tempting human beings and turning them away from God. God tells him that any humans who follow him will join him in the fire of hell at judgment day, but that Iblis will have no power over all mankind except who wants to follow Iblis.[citation needed] This story is cited multiple times in the Qur'an for different reasons.

Islamic literature presents Iblis as God worshipping and very pious until he refused to prostrate to Adam due to his jealousy and pride. Iblis is also considered to a type of supernatural being known as the Jinn, instead of an angel, who were made out of smokeless fire and created before humankind.[91][92][93][94]

Rudolf Steiner's writings, which formed the basis for Anthroposophy, characterised Lucifer as a spiritual opposite to Ahriman, with Christ between the two forces, mediating a balanced path for humanity. Lucifer represents an intellectual, imaginative, delusional, otherworldly force which might be associated with visions, subjectivity, psychosis and fantasy. He associated Lucifer with the religious/philosophical cultures of Egypt, Rome and Greece. Steiner believed that Lucifer, as a supersensible Being, had incarnated in China about 3000 years before the birth of Christ.

Luciferianism is a belief system that venerates the essential characteristics that are affixed to Lucifer. The tradition, influenced by Gnosticism, usually reveres Lucifer not as the devil, but as a liberator, a guardian or guiding spirit or even the true god as opposed to Jehovah.[96]

In Anton LaVey's The Satanic Bible, Lucifer is one of the four crown princes of hell, particularly that of the East, the 'lord of the air', and is called the bringer of light, the morning star, intellectualism, and enlightenment. The title 'lord of the air' is based upon Ephesians 2:2, which uses the phrase 'prince of the power of the air' to refer to the pagan god Zeus, but that phrase later became conflated with Satan.

Author Michael W. Ford has written on Lucifer as a "mask" of the adversary, a motivator and illuminating force of the mind and subconscious.[98]

Fourier Series


A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. Examples of successive approximations to common functions using Fourier series are illustrated above.

In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved in the case of a single sinusoid, the solution for an arbitrary function is immediately available by expressing the original function as a Fourier series and then plugging in the solution for each sinusoidal component. In some special cases where the Fourier series can be summed in closed form, this technique can even yield analytic solutions.

Any set of functions that form a complete orthogonal system have a corresponding generalized Fourier series analogous to the Fourier series. For example, using orthogonality of the roots of a Bessel function of the first kind gives a so-called Fourier-Bessel series.

The computation of the (usual) Fourier series is based on the integral identities


for m,n!=0, where delta_(mn) is the Kronecker delta.

Using the method for a generalized Fourier series, the usual Fourier series involving sines and cosines is obtained by taking f_1(x)=cosx and f_2(x)=sinx. Since these functions form a complete orthogonal system over [-pi,pi], the Fourier series of a function f(x) is given by




and n=1, 2, 3, .... Note that the coefficient of the constant term a_0 has been written in a special form compared to the general form for a generalized Fourier series in order to preserve symmetry with the definitions of a_n and b_n.

The Fourier cosine coefficient a_n and sine coefficient b_n are implemented in the Wolfram Language as FourierCosCoefficient[expr, t, n] and FourierSinCoefficient[expr, t, n], respectively.

A Fourier series converges to the function f^_ (equal to the original function at points of continuity or to the average of the two limits at points of discontinuity)

 f^_={1/2[lim_(x->x_0^-)f(x)+lim_(x->x_0^+)f(x)]   for -pi<x_0<pi; 1/2[lim_(x->-pi^+)f(x)+lim_(x->pi_-)f(x)]   for x_0=-pi,pi

if the function satisfies so-called Dirichlet boundary conditions. Dini's test gives a condition for the convergence of Fourier series.


As a result, near points of discontinuity, a "ringing" known as the Gibbs phenomenon, illustrated above, can occur.

For a function f(x) periodic on an interval [-L,L] instead of [-pi,pi], a simple change of variables can be used to transform the interval of integration from [-pi,pi] to [-L,L]. Let


Solving for x^' gives x^'=Lx/pi, and plugging this in gives




Similarly, the function is instead defined on the interval [0,2L], the above equations simply become


In fact, for f(x) periodic with period 2L, any interval (x_0,x_0+2L) can be used, with the choice being one of convenience or personal preference (Arfken 1985, p. 769).

The coefficients for Fourier series expansions of a few common functions are given in Beyer (1987, pp. 411-412) and Byerly (1959, p. 51). One of the most common functions usually analyzed by this technique is the square wave. The Fourier series for a few common functions are summarized in the table below.

functionf(x)Fourier series
Fourier series--sawtooth wavex/(2L)1/2-1/pisum_(n=1)^(infty)1/nsin((npix)/L)
Fourier series--square wave2[H(x/L)-H(x/L-1)]-14/pisum_(n=1,3,5,...)^(infty)1/nsin((npix)/L)
Fourier series--triangle waveT(x)8/(pi^2)sum_(n=1,3,5,...)^(infty)((-1)^((n-1)/2))/(n^2)sin((npix)/L)

If a function is even so that f(x)=f(-x), then f(x)sin(nx) is odd. (This follows since sin(nx) is odd and an even function times an odd function is an odd function.) Therefore, b_n=0 for all n. Similarly, if a function is odd so that f(x)=-f(-x), then f(x)cos(nx) is odd. (This follows since cos(nx) is even and an even function times an odd function is an odd function.) Therefore, a_n=0 for all n.

The notion of a Fourier series can also be extended to complex coefficients. Consider a real-valued function f(x). Write


Now examine




The coefficients can be expressed in terms of those in the Fourier series

={1/(2pi)int_(-pi)^pif(x)[cos(nx)+isin(|n|x)]dx n<0; 1/(2pi)int_(-pi)^pif(x)dx n=0; 1/(2pi)int_(-pi)^pif(x)[cos(nx)-isin(nx)]dx n>0
={1/2(a_n+ib_n) for n<0; 1/2a_0 for n=0; 1/2(a_n-ib_n) for n>0.

For a function periodic in [-L/2,L/2], these become


These equations are the basis for the extremely important Fourier transform, which is obtained by transforming A_n from a discrete variable to a continuous one as the length L->infty.

The complex Fourier coefficient is implemented in the Wolfram Language as FourierCoefficient[expr, t, n].

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