Mills College Mills College is an independent liberal arts and sciences college in the San Francisco Bay Area. Originally founded in 1852 as a young ladies' seminary in Benicia, California, Mills became the first women's college west of the Rockies. Currently, Mills is an undergraduate women's college in Oakland, California, with graduate programs for women and men. In 2013, U.S. History[edit] Built in 1871, Mills Hall originally housed the entire College. Mills College was initially founded as the Young Ladies Seminary at Benicia in 1852. On May 3, 1990, the Trustees announced that they had voted to admit male students.[11] This decision led to a two-week student and staff strike, accompanied by numerous displays of non-violent protests by the students.[12][13] At one point, nearly 300 students blockaded the administrative offices and boycotted classes.[14] On May 18, the Trustees met again to reconsider the decision, leading finally to a reversal of the vote.[15][16] Academics[edit] In 2013, U.S.
WebCT NOTE: Blackboard Learn has replaced WebCT as the University-wide course management system supported by ITS and CELT staff.* Instructors and designers: The deadline to move courses and content from WebCT to Blackboard Learn was July 15, 2012. However, if you missed the deadline, for a limited time ISU Bb Learn support staff may still be able to help. Please use this form to request access to old WebCT courses. Remember: If there is any chance you may want to use part or all of any WebCT courses at any time in the future, you MUST MOVE your courses—or course content—to Blackboard Learn. *The phase-out of WebCT at ISU was necessary because the product will no longer be supported by Blackboard, Inc., which acquired WebCT in 2005 and has committed itself to incorporating the best features and functionality of WebCT into upcoming updates of the Blackboard Learn platform.
Khan Academy Santa Fe College | Gainesville, FL Liouville's theorem (complex analysis In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f(z)| ≤ M for all z in C is constant. The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits at least two complex numbers must be constant. The theorem follows from the fact that holomorphic functions are analytic. If f is an entire function, it can be represented by its Taylor series about 0: where (by Cauchy's integral formula) and Cr is the circle about 0 of radius r > 0. where in the second inequality we have used the fact that |z|=r on the circle Cr. There is a short proof of the fundamental theorem of algebra based upon Liouville's theorem. Suppose that f is entire and |f(z)| is less than or equal to M|z|, for M a positive real number. where I is the value of the remaining integral. So, if k > n,
Cauchy's integral formula In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result denied in real analysis. Theorem[edit] where the contour integral is taken counter-clockwise. The proof of this statement uses the Cauchy integral theorem and similarly only requires f to be complex differentiable. This formula is sometimes referred to as Cauchy's differentiation formula. Proof sketch[edit] over any circle C centered at a. where 0 ≤ t ≤ 2π and ε is the radius of the circle. Letting ε → 0 gives the desired estimate Example[edit] Consider the function where and now
Differentiable function A differentiable function Differentiable functions can be locally approximated by linear functions. More generally, if x0 is a point in the domain of a function f, then f is said to be differentiable at x0 if the derivative f′(x0) exists. This means that the graph of f has a non-vertical tangent line at the point (x0, f(x0)). The function f may also be called locally linear at x0, as it can be well approximated by a linear function near this point. Differentiability and continuity[edit] If f is differentiable at a point x0, then f must also be continuous at x0. Most functions which occur in practice have derivatives at all points or at almost every point. Differentiability classes[edit] A function f is said to be continuously differentiable if the derivative f'(x) exists, and is itself a continuous function. is differentiable at 0, since exists. Sometimes continuously differentiable functions are said to be of class C1. Differentiability in higher dimensions[edit] See also[edit]
Complex analysis Murray R. Spiegel described complex analysis as "one of the most beautiful as well as useful branches of Mathematics". Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics. History[edit] Complex analysis is one of the classical branches in mathematics with roots in the 19th century and just prior. Complex functions[edit] For any complex function, both the independent variable and the dependent variable may be separated into real and imaginary parts: and where are real-valued functions. In other words, the components of the function f(z), can be interpreted as real-valued functions of the two real variables, x and y. Holomorphic functions[edit] See also: analytic function, holomorphic sheaf and vector bundles. Major results[edit]
Cauchy's integral theorem In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if two different paths connect the same two points, and a function is holomorphic everywhere "in between" the two paths, then the two path integrals of the function will be the same. Statement of theorem[edit] The theorem is usually formulated for closed paths as follows: let U be an open subset of C which is simply connected, let f : U → C be a holomorphic function, and let be a rectifiable path in U whose start point is equal to its end point. Then Discussion[edit] As was shown by Goursat, Cauchy's integral theorem can be proven assuming only that the complex derivative f '(z) exists everywhere in U. qualifies. which traces out the unit circle, and then the path integral is not defined (and certainly not holomorphic) at . See e.g.