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Source: http://math.uww.edu/~mcfarlat/s-prob.htm
More: http://people.hofstra.edu/stefan_Waner/realworld/tutorialsf4/unit4_3.html
In mathematics, the Karush–Kuhn–Tucker conditions (also known as the Kuhn-Tucker or the KKT conditions) are necessary for a solution in nonlinear programming to be optimal, provided some regularity conditions are satisfied. It is a generalization of the method of Lagrange multipliers to inequality constraints. The conditions are named for William Karush, Harold W. Kuhn, and Albert W. Tucker.
Let us consider the following nonlinear optimization problem:
where 
is the function to be minimized, 
(
) are the inequality constraints and 
(
) are the equality constraints, and 
and 
are the number of inequality and equality constraints, respectively.
The necessary conditions for this general equality-inequality constrained problem were first published in the Masters thesis of William Karush, although they only became renowned after a seminal conference paper by Harold W. Kuhn and Albert W. Tucker.
Necessary conditions
Suppose that the objective function, i.e., the function to be minimized, is 
and the constraint functions are 
and 
. Further, suppose they are continuously differentiable at a point 
. If is a local minimum that satisfies some regularity conditions, then there exist constants 
) and 
() such that
Stationarity:
,
Primal feasibility:
Dual feasibility:
() .
Complementary slackness:
for all .
Regularity conditions (or constraint qualifications)
In order for a minimum point be KKT, it should satisfy some regularity condition, the most used ones are listed below:
.. is constant. then it is positive-linear dependent at a vicinity of . (
) is positive-linear dependent if there exists 
not all zero such that 
) with associated multipliers 
for equalities and 
for inequalities than it doesn’t exist a sequence 
such that: 
therefore 
and 
thus 
.
such that 
and 
for all 
active in .
and 
are affine functions, then no other condition is needed to assure that the minimum point is KKT. It can be shown that
LICQ⇒MFCQ⇒CPLD⇒QNCQ,
LICQ⇒CRCQ⇒CPLD⇒QNCQ
(and the converses are not true), although MFCQ is not equivalent to CRCQ. In practice weaker constraint qualifications are preferred since they provide stronger optimality conditions.
Sufficient conditions
In some cases, the necessary conditions are also sufficient for optimality. This is the case when the objective function and the inequality constraints 
are continuously differentiable convex functions and the equality constraints hi are affine functions. It was shown by Martin in 1985 that the broader class of functions in which KKT conditions guarantees global optimality are the so called invex functions. So if equality constraints are affine functions, inequality constraints and the objective function are continuously differentiable invex functions then KKT conditions are sufficient for global optimality.
Value function
If we reconsider the optimization problem as a maximization problem with constant inequality constraints,
The value function is defined as:
(So the domain of 
is 
.) Given this definition, each coefficient, , is the rate at which the value function increases as ai increases. Thus if each ai is interpreted as a resource constraint, the coefficients tell you how much increasing a resource will increase the optimum value of our function . This interpretation is especially important in economics and is used, for instance, in utility maximization problems.
Source
In numerical analysis, the Runge–Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations.
The common fourth-order Runge–Kutta method
One member of the family of Runge–Kutta methods is so commonly used that it is often referred to as “RK4″ or simply as “the Runge–Kutta method”. Let an initial value problem be specified as follows.
, 
with 
.
We equally split the interval ![[a,b]](http://alt1.mathlinks.ro/latexrender/pictures/1/4/7/1475c39442659c6febe03b2e1809dbde6ec5fb5a.gif)
into 
subintervals by the following interior points
.
Of course, 
. The number 
, which will be denoted by 
, is called the step size. The value of 
at step is usually denoted by 
, that is, 
. Therefore, the initial value problem can be rewritten as , where and with 
. The Runge–Kutta 4 method tells us that, having the value of at 
, we can compute the value of at 
by using the following scheme
Thus, for each step, we need to compute all four 
, 
, constants.
The Runge–Kutta 4 method for a system of ODEs
Now we investigate the question in the title of this entry. An th-order system of 1st initial value problems has the form
for , with the initial conditions
, 
,…,
.
Let be 
. We denote by 
the following number
At each step 
, having all we can compute 
for all 
. To do this, for each , we need to compute all 
, 
. More precisely, we need to compute
As can be seen, at step 
and for each 
, all the values 
, with must be computed before any of the expressions 
.
Example
We now consider the following example
Therefore, at the step , we can compute 
and 
by using the following scheme
In linear algebra, a 
decomposition (also called a factorization) of a matrix is a decomposition of the matrix into an orthogonal and a right triangular matrix. decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the algorithm.
Square matrix
A QR decomposition of a real square matrix 
is a decomposition of as 
where 
is an orthogonal matrix (meaning that 
) and 
is an upper triangular matrix (also called right triangular matrix). This generalizes to a complex square matrix and a unitary matrix . If is nonsingular, then this factorization is unique if we require that the diagonal elements of are positive.
Rectangular matrix
More generally, we can factor a complex 
matrix , with 
, as the product of an 
unitary matrix and an upper triangular matrix . As the bottom 
rows of an upper triangular matrix consist entirely of zeroes, it is often useful to partition , or both and :
where 
is an 
upper triangular matrix, 
is , 
is 
, and and both have orthogonal columns.
Golub & Van Loan call 
the thin factorization of . If is of full rank 
and we require that the diagonal elements of are positive then and are unique, but in general is not. is then equal to the upper triangular factor of the Cholesky decomposition of 
(
if is real).
, 
and 
decompositions
Analogously, we can define , , and decompositions, with 
being a lower triangular matrix.
How to solve the Linear Least Square problems
Having the decomposition, that is, 
, one can solve the Least Square Problem 
where a matrix, 
and 
with 
. To be precise, by mean of , we see that is an matrix and is an matrix. Therefore, we can assume both and to be
![Q = \left( {\begin{array}{*{20}{c}} {{{\left[ {{Q_1}} \right]}_{m \times n}}} & {{{\left[ {{Q_2}} \right]}_{m \times \...](http://alt2.mathlinks.ro/latexrender/pictures/d/1/7/d177fe503a9df3ceca3374824c92382e0a3a61d6.gif)
Now equation is equivalent to 
. Thus 
.
Computing the decomposition
There are several methods for actually computing the QR decomposition, such as by means of the Gram–Schmidt process, Householder transformations, or Givens rotations. Each has a number of advantages and disadvantages. In this section, we mainly consider the Gram–Schmidt process. More precisely, consider the Gram–Schmidt process, with the vectors to be considered in the process as the columns of the matrix 
. Assume that after finishing the Gram–Schmidt process, we obtain 
. Therefore, we put
and thus,
.
Example
Let us consider the case when
Then
From 
, we obtain 
from 
, that is,
To obtain 
, we firstly project 
onto , then we subtract this result from . More precisely,
where
which yields
Thus
and
Finally,
In Maple, you can practice as shown below.
Linear Least Square Problem (linear least squares), or ordinary least squares (OLS), is an important computational problem, that arises primarily in applications when it is desired to fit a linear mathematical model to measurements obtained from experiments. The goals of linear least squares are to extract predictions from the measurements and to reduce the effect of measurement errors. Mathematically, it can be stated as the problem of finding an approximate solution to an overdetermined system of linear equations. In statistics, it corresponds to the maximum likelihood estimate for a linear regression with normally distributed error.
Linear least square problems admit a closed-form solution, in contrast to non-linear least squares problems, which often have to be solved by an iterative procedure.
Motivational example
As a result of an experiment, four 
data points were obtained, 
, 
, 
, and 
(shown in red in the picture on the right). It is desired to find a line 
that fits “best” these four points. In other words, we would like to find the numbers 
and 
that approximately solve the overdetermined linear system
of four equations in two unknowns in some “best” sense.
The least squares approach to solving this problem is to try to make as small as possible the sum of squares of “errors” between the right- and left-hand sides of these equations, that is, to find the minimum of the function
![S(\beta_1, \beta_2)= \left[6-(\beta_1+1\beta_2)\right]^2 +\left[5-(\beta_1+2\beta_2) \right]^2 +\left[7-(\beta_1 + 3\beta_2)\...](http://alt1.mathlinks.ro/latexrender/pictures/8/1/9/81906c08b73f52943931ea68a297e3ac824946b9.gif)
The minimum is determined by calculating the partial derivatives of 
in respect to and and setting them to zero. This results in a system of two equations in two unknowns, called the normal equations, which, when solved, gives the solution
, 
and the equation 
of the line of best fit. The residuals, that is, the discrepancies between the values from the experiment and the values calculated using the line of best fit are then found to be 
, 
, 
, and 
(see the picture on the right). The minimum value of the sum of squares is
.
The general problem
Consider an overdetermined system (there are more equations than unknowns)
of linear equations in unknown coefficients, 
, with , written in matrix form as 
where
Such a system usually has no solution, and the goal is then to find the coefficients 
which fit the equations “best”, in the sense of solving the quadratic minimization problem
A justification for choosing this criterion is given in properties below. This minimization problem has a unique solution, provided that the n columns of the matrix 
are linearly independent, given by solving the normal equations
Singular value decomposition
In linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with many applications in signal processing and statistics. Applications which employ the SVD include computing the pseudoinverse, least squares fitting of data, matrix approximation, and determining the rank, range and null space of a matrix.
Suppose 
is an -by- matrix whose entries come from the field 
, which is either the field of real numbers or the field of complex numbers. Then there exists a factorization of the form
where 
is an -by- unitary matrix over , the matrix 
is -by- diagonal matrix with nonnegative real numbers on the diagonal, and 
denotes the conjugate transpose of , an -by- unitary matrix over . Such a factorization is called a singular-value decomposition of . A common convention is to order the diagonal entries 
in non-increasing fashion. In this case, the diagonal matrix is uniquely determined by (though the matrices and are not). The diagonal entries of are known as the singular values of .
In practice, matrix consists of eigenvectors of matrix 
while matrix consists of eigenvectors of matrix 
. Finally, matrix is the square root of a diagonal matrix forming by eigenvalues of either matrix or matrix .
Example
Let us consider the case when
Having yields
By simple calculation, one has
being eigenvalues of matrix (in the descending form), which helps us to write down
To construct matrix , one has to find eigenvectors corresponding to eignvalues 
and 
. For the first one, that is, , one has the following linear system
which implies, for example,
Similar to , one has its corresponding eigenvector as follows
Thus
The numerical solution of is
which gives us
Having we can compute by the following formula 
or you can perform the same routine as above. That is
.
It is easy to see that
Having a SVD result, that is
one can see that the solution of can be computed by 
Note that matrices and are not unique. Using Maple software, you can compute SVD as shown below.
Useful links
Suppose that is an analytic function which is defined in the upper half-disk ![\{|z|^2<1,I[z]>0\}](http://alt1.mathlinks.ro/latexrender/pictures/0/4/3/043fe893c3726bac6c67116e0f9b6337a4cf826a.gif)
. Further suppose that extends to a continuous function on the real axis, and takes on real values on the real axis. Then can be extended to an analytic function on the whole disk by the formula
and the values for 
reflected across the real axis are the reflections of 
across the real axis. It is easy to check that the above function is complex differentiable in the interior of the lower half-disk. What is remarkable is that the resulting function must be analytic along the real axis as well, despite no assumptions of differentiability.
This is called the Schwarz reflection principle, and is sometimes also known as Schwarz’s symmetric principle (Needham 2000, p. 257). The diagram above shows the reflection principle applied to a function f defined for the upper half-disk (left figure; red) and its image (right figure; red). The function is real on the real axis, so it is possible to extend the function to the reflected domain (left and right figures; pink).
For the reflected function to be continuous, it is necessary for the values at the boundary to be continuous and to fall on the line being reflected. The reflection principle also applies in the generality of reflecting along any line, not just the real axis, in which case the function has to take values along a line in the range. In fact, any arc which has a neighborhood biholomorphic to a straight line can be reflected across. The basic example is the boundary of the unit circle which is mapped to the real axis by 
.
The reflection principle can also be used to reflect a harmonic function which extends continuously to the zero function on its boundary. In this case, for negative , defining
extends 
to a harmonic function on the reflected domain. Again note that it is necessary for 
. This result provides a way of extending a harmonic function from a given open set to a larger open set (Krantz 1999, p. 95).
Source:
Several applications
be a function meromorphic in the closed unit disk 
. Suppose that 
is constant on the unit circle 
. Prove that must be a rational function.Proof. If 
on 
, then f = 0 and the statement is true. W.L.O.G suppose 
on , i.e. 
and suppose that 
is holomorphic as a function on 
to the Riemann sphere 
, and 
is continuous. Since 
is compact, so is 
and thus closed in . In particular 
. Furthermore, since non-constant holomorphic mappings are open, 
is open. In conclusion
.
Therefore either 
or 
and, in both cases, 
.
Suppose (the other case is treated similarly). Now by
i.e. reflection along , we have a, in fact, the holomorphic continuation of to by the Schwarz reflection principle (if you’re not familiar with this: The idea is that this continuation is clearly holomorphic in 
; on use Morera’s theorem). We know not that 
is holomorphic. Thus has to be a rational function.
Comments: In the above solution, we can construct
to be a holomorphic function by mean of the Schwarz Reflection Principle. Roughtly speaking, we first go back to the domain 
by the mapping 
. Then we compute its image under the function . Then we go forward by the mapping
which equals to
.
The above complex conjugate operation reflects the word “reflection ” in this principle. Similar to the above question, we have the following
be holomorphic in the unit disk 
, continuous in and 
whenever . Prove that is a rational function.Proof. If has infinite many zeros, by the isolatedness of the zeros of holomorphis functions, the zeros must have limit points on the boundary of the unit disk, otherwise, 
. But it will violate the fact that is continuous in 
and whenever . Hence has only finite zeros in the unit disk. Denote all these zeros by 
, multiple zeros being repeated, and define,
Then 
is holomorphic in 
, continuous in and 
whenever . By Maximum Modulus Principle, 
in . Since 
has no zero in , 
is also holomorphic in , continuous in and 
whenever . Application of the Maximum Modulus Principle to yields 
in . Hence holds in , which implies 
with 
a real number. So we obtain
which completes the proof.
Comments: It doesn’t have to just sound like one. Basically, if has a pole in 
you should multiply by the conformal disc automorphism 
and divide by it if has a zero in . The motivation is basically just that I knew it 
. The standard way to construct an analytic function in the disc with zeros in 
is to multiply together infinitely many factors
.
This is called a Blaschke product. There are factorization theorems saying that analytic functions belonging to certain growth classes (the Smirnov class f.ex.) can be factored into products of a Blaschke product (describing the zeros), a singular inner function (describing the angle of ), and an outer function (describing the growth). With the very nice function described in the problem I knew that it would have no growth and no inner factor either (they can’t be continuous on the circle), so essentially I just knew the solution to the problem. Although I guess the solution can be easily guessed if products of Möbius transforms of the type above are familiar to you as exactly the rational functions that have constant modulus on the unit circle.
In this small topic, I will show several traditional results regarding to the perturbation in matrix algebra. This comes from the study of the stability of solving system of linear equations in Numerical Analysis.
Let us consider the following matrix-form linear system . We split into three cases
If 
is a small perturbation on , then 
is a small perturbation on in the sense that the following system 
holds true. This together with the fact that 
gives 
. By some simple calculation, one has
Thus
.If 
is a small perturbation on , then is a small perturbation on in the sense that the following system 
holds true. This together with the fact that gives 
. By some simple calculation, one has
Thus
which implies that
and .If is a small perturbation on and is a small perturbation on , then is a small perturbation on in the sense that the following system 
holds true. This together with the fact that gives 
. By some simple calculation, one has
which implies that
Thus,
In the literature, the expression 
is called the condition number of the matrix , denoted by 
. It is easy to see that 
. If is small then the system is called to be well-posed. Alternatively, in the case when 
, the system is ill-posed.
Truncation error or local truncation error is error made by numerical algorithms that arises from taking finite number of steps in computation. It is present even with infinite-precision arithmetic, because it is caused by truncation of the infinite Taylor series to form the algorithm.
Use of arbitrarily small steps in numerical computation is prevented by round-off error, which are the consequence of using finite precision floating point numbers on computers.
Let 
represent the difference equation approximating the PDE at the 
-th mesh point, with exact solution 
. For example,
represent the difference equation approximating the following PDE 
at the -th mesh point. If is replaced by at the mesh points of the 
E, where is the exact solution of the PDES, the value of 
is called the local truncation error 
at the -th mesh point. clearly measures the amount by which the exact solution values of the PDE at the mesh points of the E do not satisfy the E at the point 
.
Continuing the above example gives us
Using Taylor expansions, it is easy to express in terms of power of and 
and partial derivatives of at . This leads us to the computation of the local truncation error.
It is sometimes possible ot approximate a parabolic or hyperbolic equation by a finite-difference scheme that is stable (i.e. limits the amplification of all the compoonents of the initial conditions), but which has a solution that converges to the solution of a different differential equation as the mesh lengths tend to zero. Such a difference scheme is said to be inconsistent or imcompatible with the PDE.
The real importance of the concept of consistency lies in a theorem by Lax which states that if a linear finite-difference equation is consistent with a properly posed linear initial-value problem then stability guarantees convergnce of to as the mesh lengths tned to zero. Consistency can be defined in either of two equivalent but slightly different ways.
The more general definition is as follows. Let 
represent the PDE in the independent variables and 
, with exact solution . Let 
represent the approximating finite-difference equation with exact solution . Let be a continuous function of and with a sufficient number of continuous derivatives to enable 
to be evaluated at the point 
.
Then the truncation error 
at the point is defined by
If 
as 
and 
, the difference equation is said to be consistent or compatible with the PDE. Most authors put 
because .
For example, let us consider the following parabolic equation
with
and
where 
and 
.
Backward Euler + second order central finite difference discretization.
We split equally the domain 
into parts and the domain 
into parts, i.e. 
-th mesh point is of the following form 
. We are now in a position to discretize the problem as follows
where 
. At 
, that is 
, one has
and at 
, that is 
, one obtains
The local truncation error and consistency
The above discretization can be rewritten as following
where with the following boundary conditions
and
Now we have
By Taylor’s expansion
and
and
Therefore,
which yields
Thus, the principle part of the local truncation error is
Hence 
Consistency.
In view of the local truncation error, one can easily see that the local truncation error is a polynomial of two variables and which implies that the method is consistent.
In mathematics, finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.
Intuitive derivation
Finite-difference methods approximate the solutions to differential equations by replacing derivative expressions with approximately equivalent difference quotients. That is, because the first derivative of a function is, by definition,
then a reasonable approximation for that derivative would be to take
for some small value of . In fact, this is the forward difference equation for the first derivative. Using this and similar formulae to replace derivative expressions in differential equations, one can approximate their solutions without the need for calculus.
Derivation from Taylor’s polynomial
Assuming the function whose derivatives are to be approximated is properly-behaved, by Taylor’s theorem,
where 
denotes the factorial of , and 
is a remainder term, denoting the difference between the Taylor polynomial of degree and the original function. Again using the first derivative of the function as an example, by Taylor’s theorem,
which, with some minor algebraic manipulation, is equivalent to
so that for 
sufficiently small,
Accuracy and order
The error in a method’s solution is defined as the difference between its approximation and the exact analytical solution. The two sources of error in finite difference methods are round-off error, the loss of precision due to computer rounding of decimal quantities, and truncation error or discretization error, the difference between the exact solution of the finite difference equation and the exact quantity assuming perfect arithmetic (that is, assuming no round-off).
To use a finite difference method to attempt to solve (or, more generally, approximate the solution to) a problem, one must first discretize the problem’s domain. This is usually done by dividing the domain into a uniform grid (see image to the right). Note that this means that finite-difference methods produce sets of discrete numerical approximations to the derivative, often in a “time-stepping” manner.
An expression of general interest is the local truncation error of a method. Typically expressed using Big-O notation, local truncation error refers to the error from a single application of a method. That is, it is the quantity 
if 
refers to the exact value and 
to the numerical approximation. The remainder term of a Taylor polynomial is convenient for analyzing the local truncation error. Using the Lagrange form of the remainder from the Taylor polynomial for 
, which is
where 
, the dominant term of the local truncation error can be discovered. For example, again using the forward-difference formula for the first derivative, knowing that 
,
and with some algebraic manipulation, this leads to
and further noting that the quantity on the left is the approximation from the finite difference method and that the quantity on the right is the exact quantity of interest plus a remainder, clearly that remainder is the local truncation error. A final expression of this example and its order is:
This means that, in this case, the local truncation error is proportional to the step size.
Explicit method
Using a forward difference at time 
and a second-order central difference for the space derivative at position 
(”FTCS”) we get the recurrence equation:
This is an explicit method for solving the one-dimensional heat equation. We can obtain 
from the other values this way:
where 
. So, knowing the values at time you can obtain the corresponding ones at time 
using this recurrence relation. 
and 
must be replaced by the boundary conditions, in this example they are both 0.
This explicit method is known to be numerically stable and convergent whenever 
. The numerical errors are proportional to the time step and the square of the space step:
Implicit method
If we use the backward difference at time and a second-order central difference for the space derivative at position (”BTCS”) we get the recurrence equation:
This is an implicit method for solving the one-dimensional heat equation.
We can obtain 
from solving a system of linear equations:
The scheme is always numerically stable and convergent but usually more numerically intensive than the explicit method as it requires solving a system of numerical equations on each time step. The errors are linear over the time step and quadratic over the space step.
Crank-Nicolson method
Finally if we use the central difference at time 
and a second-order central difference for the space derivative at position (”CTCS”) we get the recurrence equation:
This formula is known as the Crank-Nicolson method. We can obtain from solving a system of linear equations:
The scheme is always numerically stable and convergent but usually more numerically intensive as it requires solving a system of numerical equations on each time step.
The errors are quadratic over the time step and formally are of the fourth degree regarding the space step:
However, near the boundaries, the error is often 
instead of 
. Usually the Crank-Nicolson scheme is the most accurate scheme for small time steps. The explicit scheme is the least accurate and can be unstable, but is also the easiest to implement and the least numerically intensive. The implicit scheme works the best for large time steps.
In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the remarkable behavior of meromorphic functions near essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati.
Formal statement of the theorem
Start with some open subset in the complex plane containing the number 
, and a function that is holomorphic on 
, but has an essential singularity at . The Casorati–Weierstrass theorem then states that
if is any neighborhood of contained in , then 
is dense in 
.
This can also be stated as follows:
for any 
and any complex number 
, there exists a complex number in with 
and 
.
Or in still more descriptive terms:
comes arbitrarily close to any complex value in every neighbourhood of .
This form of the theorem also applies if is only meromorphic. The theorem is considerably strengthened by Picard’s great theorem, which states, in the notation above, that assumes every complex value, with one possible exception, infinitely often on .
Examples
The function 
has an essential singularity at 
, but the function 
does not (it has a pole at 0). Consider the function
This function has the following Laurent series about the essential singular point at :
Because 
exists for all points 
we know that is analytic in the neighborhood of 
. Hence it is an isolated singularity, as well as being and essential singularity. Using a change of variable to polar coordinates 
our function, becomes:
Taking the absolute value of both sides:
Thus, for values of 
such that 
, we have 
as 
, and for 
, 
as .
Consider what happens, for example when takes values on a circle of diameter 
tangent to the imaginary axis. This circle is given by 
. Then,
![f(z) = e^{R} \left[ \cos \left( R\tan \theta \right) - i \sin \left( R\tan \theta \right) \right]](http://alt1.mathlinks.ro/latexrender/pictures/2/8/3/283621029570623e264c81b0caa629b93d38f8f6.gif)
and
Thus,
may take any positive value other than zero by the appropriate choice of . As 
on the circle, 
with fixed. So this part of the equation:
![\left[ \cos \left( R \tan \theta \right) - i \sin \left( R \tan \theta \right) \right]](http://alt2.mathlinks.ro/latexrender/pictures/c/b/5/cb552371dea96098fdd6fa81977c3bf928b36747.gif)
takes on all values on the unit circle infinitely often. Hence takes on all the value of every number in the complex plane except for zero infinitely often.
Proof of the theorem
A short proof of the theorem is as follows: Take as given that function is meromorphic on some punctured neighborhood 
, and that is an essential singularity. Assume by way of contradiction that some value exists that the function can never get close to; that is: assume that there is some complex value and some such that 
for all in at which is defined.
Then the new function:
must be holomorphic on , with zeroes at the poles of , and bounded by 
. It can therefore be analytically continued (or continuously extended, or holomorphically extended) to all of by Riemann’s analytic continuation theorem. So the original function can be expressed in terms of :
for all arguments in . Consider the two possible cases for 
If the limit is 0, then has a pole at . If the limit is not 0, then is a removable singularity of . Both possibilities contradict the assumption that the the point is an essential singularity of the function . Hence the assumption is false and the theorem holds.
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