Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

i. e., z is a vector orthogonal to the vector v ( Indeed, z is the projection of u onto the plane orthogonal to v .) We can thus apply the Pythagorean theorem to

We can prove the cancellation law easily using Euclid's lemma, which generally states that if an integer b divides a product rs ( where r and s are integers ), and b is relatively prime to r, then b must divide s. Indeed, the equation

" Indeed, Gehry himself had made the comparison, " We started collecting pictures of Stratocasters, bringing in guitar bodies, drawing on those shapes in developing our ideas.

Indeed, We declared war on America and Britain out of Our sincere desire to ensure Japan's self-preservation and the stabilization of East Asia, it being far from Our thought either to infringe upon the sovereignty of other nations or to embark upon territorial aggrandizement.

Indeed, " Anything For You " became the top-selling single in Jamaica in 1995 and " Si Wi Dem Nuh Know We " also reached the number one slot in Jamaica.

Pope Pius XI ( 1857-1939 ) was so impressed by the courage of the Oblates that he referred to them as “ Specialists in the most difficult missions of the Church ” Indeed, regarding their ministry the Oblates declare :” We fulfill our task in healing the world by understanding its evolutionary character, by critically engaging its contemporary spirit, and by meeting its new needs in new ways.

Indeed the band had written to Evans, however Gary Stringer explained it differently, “ We don ’ t want people to think that ’ s all we ’ re about ” and went on “... he ’ s gone on telly and said we storm offstage whenever someone in the crowd sings that version of it.

Indeed Biko rejected the " practicality " charge as an example of the compromises that hindered and delayed black liberation, saying in 1977: " We have been successful to the extent that we have diminished the element of fear in the minds of black people.

Indeed, as is well known to those who study the history of science, it must be recognized on the one hand that the Roman Pontiffs and the Catholic Church have always fostered the research of the learned in the experimental field as well, and on the other hand that such research has opened up the way to the defense of the deposit of supernatural truths entrusted to the Church .... We promise again that it is our strongly-held intention, that the ' Pontifical Academicians ' through their work and our Institution, work ever more and ever more effectively for the progress of the sciences.

Indeed on 3 July 1935, the KKE newspaper, Rizospastis, reported a statement issued by the IMRO ( United ) organisation in Edesa ( Voden ) and signed by G. Slavos: "... We Macedonians also insist on not being called Bulgarians, for we are neither Bulgarians, nor Serbs, nor Greeks, but Macedonians.

Indeed, he concluded that `` geographical conditions have exercised a great, possibly a preponderating, influence over man's destiny ''.

Indeed, again and again, the space merchants confirm the prediction of the humanists that the conditioners and behavioral scientists, once they have seen through human nature, will have nothing except their impulses and desires to guide them.

Indeed, it has only been a matter of the last few years that reputable schools of art have granted degrees at all.

Indeed, we should say, on the contrary, that the accident of our later discovery made no difference whatever to the badness of the animal's pain, that it would have been every whit as bad whether a chance passer-by happened later to discover the body and feel repugnance or not.

Indeed, in some periods of our history and in some neighborhoods the job opportunities have been so good that undoubtedly a great many boys who were potential members of the professions quit school at an early age and went to work.

Indeed the extensive use of imported non-Greeks (" barbarians ") as chattel slaves seems to have been an Athenian development.

Indeed the urban areas of the country suffering heavily from unemployment, which might have been expected to respond the most to the radical economic policies of the Liberals, instead gave the party its worst results.

Indeed, bongos are the only Tragelaphid in which both sexes have horns.

Indeed, some Bayesians have argued the prior state of knowledge defines the ( unique ) prior probability-distribution for " regular " statistical problems ; cf.

Indeed, methods for constructing " objective " ( alternatively, " default " or " ignorance ") priors have been developed by avowed subjective ( or " personal ") Bayesians like James Berger ( Duke University ) and José-Miguel Bernardo ( Universitat de València ), simply because such priors are needed for Bayesian practice, particularly in science.

Indeed, even after 1790 Mozart writes about " the rehearsal ", with the implication that his concerts would have only one.

Indeed, scholars have referred to the presence or absence of such features as one source of evidence for dating the remains.

Indeed, in those cases where no clear consensus exists on a given norm, the drafting of criminal law by the group in power to prohibit the behaviour of another group may seem to some observers an improper limitation of the second group's freedom, and the ordinary members of society have less respect for the law or laws in general — whether the authorities actually enforce the disputed law or not.

Indeed, despite everything, the majority of natural-law theorists have accepted the idea of enforcing the prevailing morality as a primary function of the law.

Indeed, many polls have indicated that it is often a favorite dish.

Indeed John Morris, the English historian who specialized in the study of the institutions of the Roman Empire and the history of Sub-Roman Britain, suggested in his book The Age of Arthur that as the descendants of Romanized Britons looked back to a golden age of peace and prosperity under Rome, the name " Camelot " of Arthurian legend may have referred to the capital of Britannia ( Camulodunum, modern Colchester ) in Roman times.

Indeed, proponents of ethical naturalism have argued that humanity needs to invest in their science of morality – although the existence of such a science is debated.

Indeed, Jenner may have been aware of Jesty's procedures and success.

Indeed, these Four Freedoms were explicitly incorporated into the preamble to the Universal Declaration of Human Rights which reads, " Whereas disregard and contempt for human rights have resulted in barbarous acts which have outraged the conscience of mankind, and the advent of a world in which human beings shall enjoy freedom of speech and belief and freedom from fear and want has been proclaimed the highest aspiration of the common people ,...."

Indeed, at the end of the 19th century scholar Ernest DeWitt Burton wrote that there could be " no reasonable doubt " that 1 John and the gospel were written by the same author, and Amos Wilder has said that, " Early Christian tradition and the great majority of modern scholars have agreed on the common authorship of these writings, even where the author has not been identified with the apostle John.

Indeed many descriptions of Hobart have used the phrase " nestled amidst the foothills ", so undulating is the landscape.

Indeed, the Zulu attacks on the British strongpoints at Rorke's Drift and at Kambula, ( both bloody defeats ) seemed to have been carried out by over-enthusiastic leaders and warriors despite contrary orders of the Zulu King, Cetshwayo.

Indeed, most of the offensive and defensive toxins that organisms have evolved for shutting down the nervous systems of predators and prey ( e. g., the venoms produced by spiders, scorpions, snakes, fish, bees, sea snails and others ) work by modulating ion channel conductance and / or kinetics.

Indeed, it is in the field of transportation that Congress has most frequently granted employers exemption from the anti-trust laws ; ;

Indeed, Class “ A ” status was granted to a number of Arab peoples who were ready for independence in the former Ottoman Empire, and only to Arab entities.

Indeed, interoperability is not taken for granted in the non-standards-based portion of the computing world.

Indeed, against Liu Ye's advice, he appointed Sun Quan the King of Wu and granted him the nine bestowments.

Indeed, in October 1981, they gave a demo tape to Peel, who granted them a Peel Session.

Indeed, on Yom Kippur, the Shofar was sounded to announce the Jubilee Year ( every 50 years, Jews were granted forgiveness, debts were forgiven, indentured Israelites were granted freedom, and the fields " shall become owned by the priests.

Indeed, the proclamation itself called for lands to be granted to British soldiers that had served in the Seven Years ' War.

Indeed the decline was so severe that in 1601 Argeleb II ( r. 1589 – 1670 ) granted a large portion of Arthedain's best farmland to Hobbit migrants, as these lands had become deserted.

Indeed, against Liu Ye's advice, he created Sun the Prince of Wu and granted him the nine bestowments.

Indeed, Charlemagne granted immunity to the convent at Hohenberg, which was later officially ratified by Louis the Pious on 9 March 837.

Indeed John Canynges and his business partner William Cheddar the Elder had taken temporary possession of William Corbet's 2 / 3rds occupancy of Alveston manor as security for their debt, and later granted it by gift to William II Canynges ( d. 1474 ) who held it from them as a " free tenant ".

Indeed the King might have granted this, partly through mercy and partly to avoid upsetting Thomas, Earl of Derby.

Indeed, when the title was granted with Ras Warq ( the right to wear a coronet ), it was higher than even the title of Ras.