Then the complex derivative of ƒ at a point z < sub > 0 </ sub > is defined by

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.

Then Goursat's theorem asserts that ƒ is analytic in an open complex domain Ω if and only if it satisfies the Cauchy – Riemann equation in the domain.

Then, using the periodic Bernoulli function P < sub > n </ sub > defined above and repeating the argument on the interval, one can obtain an expression of ƒ ( 1 ).

Then ƒ is a homomorphism of rings, since ƒ preserves both addition:

Then ƒ is a homomorphism of groups, since it preserves multiplication:

Then for arbitrary ε > 0 there is an embedding ( or immersion ) ƒ < sub > ε </ sub >: M < sup > m </ sup > → R < sup > n </ sup > which is

Then the partial fraction decomposition of ƒ ( x ) is the following:

Then, the value ƒ ( x < sub > 0 </ sub >) will be larger than other values ƒ ( x ).

Then for any r between p and q we have that F is dense in, that Tƒ is in for any ƒ in F and that T is bounded in the norm.

Then ƒ is equal to a finite Blaschke product

Then ƒ ( z ) can be expanded in terms of polynomials A < sub > n </ sub > as follows:

Then for a function ƒ on S < sup > n − 1 </ sup >, the spherical Laplacian is defined by

Then, if ƒ ( x < sub > 0 </ sub >) is not equal to, x < sub > 0 </ sub > is called a removable discontinuity.

Then ƒ satisfies () precisely when it is a conformal transformation from D equipped with this metric to the domain D ′ equipped with the standard Euclidean metric.

Then he would get to his feet, as though rising in honor of his own remarkable powers, and say almost invariably, `` Gentlemen, this is an amazing story!!

Then, Jesus indicated that God's forgiveness is unlimited.

Certainly, the meaning is clearer to one who is not familiar with Biblical teachings, in the New English Bible which reads: `` Then Jesus arrived at Jordan from Galilee, and he came to John to be baptized by him.

Then it added: `` It is not possible to determine how extensive these ill effects will be -- nor how many people will be affected ''.

Then the words fell into a pattern: `` Mollie the Mutton is scratching her nose, Scratching her nose in the rain.

Then he thought of Aaron Blaustein standing in his rich house saying: `` God is tired of taking the blame.

Then it is marked on the inside where it comes in contact with the transom, frames, keelson and all the battens.

Then it is replaced and fastened.

Then the chines are rounded off and the bottom is rough-sanded in preparation.

Then, a group of eggs is deposited in a cavity in the beebread loaf and the egg compartment is closed.

Then there is a diagonalizable operator D on V and a nilpotent operator N in V such that ( A ) Af, ( b ) Af.

Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

Then, too, the utmost clinical flexibility is necessary in judiciously combining carefully timed family-oriented home visits, single and group office interviews, and appropriate telephone follow-up calls, if the worker is to be genuinely accessible and if the predicted unhealthy outcome is to be actually averted in accordance with the principles of preventive intervention.

Then the editorial added prophetically: `` how far they may reach in Asia is yet undetermined, but they fall far short of our dreams of the war conferences ''.

Then she catapults into `` everything and everybody '', putting particular violence on `` everybody '', indicating to the linguist that this is a spot to flag -- that is, it is not congruent to the patient's general style of speech up to this point.

Then comes the time when the last wire is removed and Susie walks out a healthier and more attractive girl than when she first went to the orthodontist.

Then, with the new affluence, there is actually a sallying forth into the wide, wide world beyond the precincts of New York.

Then, if the middle number is activated to its greatest potential in terms of this square, through multiplying it by the highest number, 9 ( which is the square of the base number ), the result is 45 ; ;

Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ ( x ) ( or more explicitly σ < sub > B </ sub >( x )) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B.

Then is separable algebraic over if and only if the matrix is invertible.

Then there is an invertible k × k matrix of functions g such that

Then I + A is invertible, and the Cayley transform

Then, to conclude on an indeterminate note: `` Nevertheless, if fallout increased substantially, or remained high for a long time, it would become far more important as a potential health hazard in this country and throughout the world ''.

Then if you change jobs you won't necessarily have to sell the farm.

Then it will be a `` fearful thing to fall into the hands of the living God '' if you have abused Him in your hands.

Then he fled, not waiting to see if she minded him or took notice of his cry.

Then, if Myra does nothing about fetching her, I'll pack her right back to her mother -- if I have to take her myself ''!!

Then, a little later, Shilkret discovered there was no one to play the brief celesta solo during the slow section, so he hastily asked Gershwin if he might play the solo ; Gershwin said he could and so he briefly participated in the actual recording.

Then another priesthood holder joins in, if available, and pronounces a " sealing " of the anointing and other words of blessing, as he feels inspired.

Then he proves the antithesis, that time has no beginning, by showing that if time had a beginning, then there must have been " empty time " out of which time arose.

Then X is reflexive if and only if each X < sub > j </ sub > is reflexive.

Then X is separable if and only if X ′ is separable.

Then g ( E ) is called the density of states if the energy spectrum is continuous.

Then X is compact if and only if X is a complete lattice ( i. e. all subsets have suprema and infima ).

Then f = u + iv is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy – Riemann equations ( 1a ) and ( 1b ) at that point.

Then, if these " digital modules " were able to build a self-sustaining business, the company would be free to use them to develop a complete computer in their Phase II.

Then the Cartesian product set D < sub > 1 </ sub > D < sub > 2 </ sub > can be made into a directed set by defining ( n < sub > 1 </ sub >, n < sub > 2 </ sub >) ≤ ( m < sub > 1 </ sub >, m < sub > 2 </ sub >) if and only if n < sub > 1 </ sub > ≤ m < sub > 1 </ sub > and n < sub > 2 </ sub > ≤ m < sub > 2 </ sub >.