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.

Indeed, even the Greenwich meridian itself is not quite what it used to be — defined by " the centre of the transit instrument at the Observatory at Greenwich ".

Indeed, MSK is a particular case of the sub-family of CPM known as continuous-phase frequency-shift keying ( CPFSK ) which is defined by a rectangular frequency pulse ( i. e. a linearly increasing phase pulse ) of one symbol-time duration ( total response signaling ).

Indeed, in the latter case, it is only defined for non-negative integer arguments.

Indeed it seems at least doubtful whether the functions of the council were ever properly defined.

Indeed, the Mandelbrot set is defined as the set of all such that is connected.

Indeed, for the f defined above, we find that

Indeed, the new spatial form of the mega-city or megalopolis, is defined by Castells as having the contradictory quality of being " globally connected and locally disconnected, physically and socially ".

Indeed relationships within the Emberizidae as defined here are uncertain with the possibility that each of the three main groups may not be all that closely related.

Indeed, beginning with the fundamental data of linguistics, the phonemic contrasts, all the elements are defined relative to one another.

Indeed, modern mathematics has been defined in a very general sense as the study of abstract structures ( by the Bourbaki group: see discussion there, at algebraic structure and also structure ).

Indeed, they are formally defined as the algebra of bounded operators leaving invariant each subspace contained in a subspace nest, that is, a set of subspaces which is totally ordered by inclusion and is also a complete lattice.

Indeed, let K be an imaginary quadratic field with class field H. Let E be an elliptic curve with complex multiplication by the integers of K, defined over H. Then the maximal abelian extension of K is generated by the x-coordinates of the points of finite order on some Weierstrass model for E over H.

Indeed, the group GL ( A )/ E ( A ) was first defined and studied by Whitehead, and is called the Whitehead group of the ring A.

Indeed, vector fields are often defined to be the derivatives of a collection of smooth curves.

Indeed, a uniformly continuous map defined on a convex set of a normed space always admits a subadditive modulus of continuity ; in particular, real-valued as a function.

Indeed, it is immediate to check that the optimal modulus of continuity defined above is subadditive if the domain of is convex: we have, for all and:

Indeed, in the early case of R v Higgins incitement was defined as being committed when one person counsels, procures or commands another to commit a crime, whether that person commits the crime.

Indeed, Central German is often defined as the area between the Appel / Apfel and the Dorp / Dorf boundaries.

Indeed, the most significant protocol defined by the SIGTRAN group is SCTP, which is used to carry PSTN signaling over IP.

Indeed, if the first mode of production was laissez-faire based capitalism, and the second mode was the model of a centrally-planned economy, then the third mode is defined neither by the motor of profit, nor by central planning: to allocate resources and make decisions, it does not use market and pricing mechanisms, or managerial commands, but instead uses social relations.

Indeed, you wouldn't live long, for the females either drive the men they've seized from neighboring islands back to their boats after exploiting them for amatory purposes, or they destroy them by revolting but ingenious methods.

Indeed, the spiral Af, with the two endpoints connected by a straight line possesses only one inscribed square.

( Indeed, the set of endings can be replaced by the name of a set of endings.

Indeed, of all that is his every man is by nature and reason and therefore by conscience obligated to regard himself as a custodian.

Indeed, when the Myth of Osiris and Isis emerged, it was said that when Osiris had been killed by Set, Osiris ' organs were given to Anubis as a gift.

Indeed, some boomerangs were not thrown at all, but were used in hand to hand combat by Indigenous Australians.

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 he is widely praised by his successors, both Labour and Conservative.

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, the energy density of some material involved in the formation of impact craters is many times higher than that generated by high explosives.

Indeed, mail armour is mentioned in the Koran as being a gift revealed by Allah to David:

Indeed, in the United States, cross-examination is seen as a core part of the entire adversarial system of justice, in that it " is the principal means by which the believability of a witness and the truth of his testimony are tested.

If φ is C < sup > k </ sup >, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided φ is continuous on the closure of D. Indeed, by the Cauchy integral formula,

Indeed, batch distillation and fractionation succeed by varying the composition of the mixture.

Indeed Hilbert would lose his " gifted pupil " Weyl to intuitionism — " Hilbert was disturbed by his former student's fascination with the ideas of Brouwer, which aroused in Hilbert the memory of Kronecker ".

" Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.

Indeed, by the age of eighteen nearly all of his closest relatives had died by war or disease.

Indeed, it was the thirty years of work done by Thomas Aquinas and himself ( 1245 – 1274 ) that allowed for the inclusion of Aristotelian study in the curriculum of Dominican schools.

Indeed the Republic of Independent Guyana, in French La République de la Guyane indépendante and commonly referred to by the name of the capital " Counani ", was created in the area which was disputed by France ( as part of French Guyana ) and Brazil in the late nineteenth century.

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, it appears the Maker of the Stars himself, by clear arguments, admonished me to call these new planets by the illustrious name of Your Highness before all others.