<p>"paffle pancake and waffle"---Google that.</p>
<p>No patent though.</p>
<p>ha, but they don't know about us. next week when they go to church on paffleday, they'll be SOOOOO suprised!!!!!</p>
<p>OMG! But we trademarked it first so we win!!</p>
<p>Monday will be Waffle
Tuesday will be Pancake
Wednesday will be Maple Syrup
Thursday will be Butter
Friday will be...umm..Grizzly Bear Day
and Saturday is...Um...</p>
<p>Holomorphic functions are central to the study of complex analysis. These functions are defined on the complex number plane C (with values that make C complex-differentiable at every point). What makes a holomorphic function so powerful is that the function can be described by its Taylor Series because the implication is that it is infinitely differentiable.</p>
<p>We will let B be an open subset of C and f. f is complex differentiable at some point z* of B if
f'(z<em>) =
lim (f(z)-f(z</em>))/(z-z<em>)
z-->z</em><br>
This limit is taken over all series of complex numbers approaching z* and for all these series, the difference quotient must approach f'(z<em>). We know through intuition that f is complex differentiable at z</em>. If we approach z* from the direction p, then our image graphs will approach f(z<em>) from the direction f'(z</em>)p. This concept is analagous to real differentiability in that it obeys the rules of differentiation (product, quotient, chain).
But the essence of the relationship between complex differentiability and real differentiability lies in the following.
f(x+iy) = a(x,y) + ib(x,y) is a holomorphic complex function, with a and b being the first partial derivatives with respect to x and y and satisfy the Cauchy-Riemann equations. But the converse is NOT true. Here is a simple converse: if a and b have partial derivatives that are continuous and satisfy the C-R equations, then f is holomorphic. A different and more complex converse is the Looman-Menchoff Theorem, which is a real pain to prove.</p>
<p>Note: If f is complex differentiable at every point z* in B, then we say that f is holomorphic on B. </p>
<p>Here are some examples of holomorphicity:
The sine, cosine, and exponential function are all holomorphic on C, as is the main branch of the logarithmic function. The square root function can be written as squareroot(z) = e^(.5*log(z)). Thus the square root function is holomorphic whenever log(z) is holomorphic.</p>
<p>It is difficult to discuss conformal mappings and holomorphicity more in depth without first touching on Shwarz Lemma, automorphisms on D and H, and the Riemann Mapping Theorem.</p>
<p>ChaosTheory?</p>
<p>saturday is french toast.
rofl. the two discussions are hilarious when put together, but i think it's kind of annoying to them. who should leave? the paffle discussion, or the people that are edumacated (sound it out, edu-ma-cated... it's a hick version of educated).</p>
<p>Yay! Our week has been created! Shall we renamed the months :D?</p>
<p>^ brilliant! must wait 12 seconds...</p>
<p>January= waffleironary
February= doughember
March= Tastebud
...</p>
<p>April= findauntjemima</p>
<p>:D </p>
<p>Alas we should rename EVERYTHING!!</p>
<p>May = Breakfasember</p>
<p>Ooooh, a french toast waffle...</p>
<p>Let's make a fraffle!</p>
<p>well, it won't be as good as the paffle, but why not?</p>
<p>hmmm, or maybe a french toast pancake....a francake!</p>
<p>...so many possibilities...</p>
<p>The CC Breakfast revolution :D /<em>take that smarty pants I ruined your thread!</em>/</p>
<p>i like poptarts! we need to do something with poptarts!!!!</p>
<p>June= illegitimatebreakfastchildmonth</p>
<p>:) /<em>francake makes me laugh</em>/</p>
<p>imapeach, what was your secret message!!!! could it be the recipe you've been working on that you don't want to be leaked out???</p>
<p>oh, i saw it. francakes aren't that funny, though they are humerous :) (though it was a letdown, after what i expected :*()</p>
<p>lol /<em>mwhahaha the invisible thread has spilled over everywhere!!</em>/</p>
<p>yes it has. and it's rather annoying to double check everything. /<em>screw you, whoever else has found the link!!!!!!</em>/</p>
<p>It's fun though!! /<em>dude not cool not cool</em>/</p>