isolated points called poles. Here, the poles are the numbers of the form for the tangent and the secant, or for the cotangent and the cosecant, where is an arbitrary integer. Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence. Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets. More precisely, defining , the th up/down number, , the th Bernoulli number, and , is the th Euler number, one has the following series expansions: Continued fraction expansion The following expansions are valid in the whole complex plane: The last one was used in the historically first proof that π is irrational. Partial fraction expansion There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: This identity can be proved with the Herglotz trick. Combining the th with the th term lead to absolutely convergent series: Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions: Infinite product expansion The following infinite product for the sine is of great importance in complex analysis: For the proof of this expansion, see Sine. From this, it can be deduced that Relationship to exponential function (Euler's formula) Euler's formula relates sine and cosine to the exponential function: This formula is commonly considered for real values of , but it remains true for all complex values. Proof: Let and One has for . The quotient rule implies thus that . Therefore, is a constant function, which equals , as This proves the formula. One has Solving this linear system in sine and cosine, one can express them in terms of the exponential function: When is real, this may be rewritten as Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity for simplifying the result. Definitions using functional equations One can also define the trigonometric functions using various functional equations. For example, the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula and the added condition In the complex plane The sine and cosine of a complex number can be expressed in terms of real