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Silverdream wrote:
Now open wide.
Colette wrote:If one assumes that the partial derivatives of a holomorphic function are continuous, the Cauchy integral theorem can be proved as a direct consequence of Green's theorem and the fact that the real and imaginary parts of must satisfy the Cauchy–Riemann equations in the region bounded by , and moreover in the open neighborhood U of this region. Cauchy provided this proof, but it was later proved by Goursat without requiring techniques from vector calculus, or the continuity of partial derivatives.
We can break the integrand , as well as the differential into their real and imaginary components:
In this case we have
By Green's theorem, we may then replace the integrals around the closed contour with an area integral throughout the domain that is enclosed by as follows:
However, being the real and imaginary parts of a function analytic in the domain , and must satisfy the Cauchy–Riemann equations there:
We therefore find that both integrands (and hence their integrals) are zero
This gives the desired result
THEREFORE, THIS THREAD IS DUMB.
Q.E.D.
samuelzz10 wrote:Colette wrote:If one assumes that the partial derivatives of a holomorphic function are continuous, the Cauchy integral theorem can be proved as a direct consequence of Green's theorem and the fact that the real and imaginary parts of must satisfy the Cauchy–Riemann equations in the region bounded by , and moreover in the open neighborhood U of this region. Cauchy provided this proof, but it was later proved by Goursat without requiring techniques from vector calculus, or the continuity of partial derivatives.
We can break the integrand , as well as the differential into their real and imaginary components:
In this case we have
By Green's theorem, we may then replace the integrals around the closed contour with an area integral throughout the domain that is enclosed by as follows:
However, being the real and imaginary parts of a function analytic in the domain , and must satisfy the Cauchy–Riemann equations there:
We therefore find that both integrands (and hence their integrals) are zero
This gives the desired result
THEREFORE, THIS THREAD IS DUMB.
Q.E.D.
who the hell cares besides colette
Apollyon wrote:Actually i looked at it for more than five seconds. The maths are easy enough but alas i am not familiar with some of the terminology or the mentioned theorems.
The Shadowscythe wrote:I sexually identify as an Apache gunship.
And my spirit animal is a Marine.
stubby wrote:Apollyon wrote:Actually i looked at it for more than five seconds. The maths are easy enough but alas i am not familiar with some of the terminology or the mentioned theorems.
Oh, memories. I used to tutor these equations in high school, back before I realized you got a lot more ladies as an artist than as a math prodigy.
stubby wrote:Apollyon wrote:Actually i looked at it for more than five seconds. The maths are easy enough but alas i am not familiar with some of the terminology or the mentioned theorems.
Oh, memories. I used to tutor these equations in high school, back before I realized you got a lot more ladies as an artist than as a math prodigy.
I used to use Cauchy to school dudes who questioned the continuity of BrikWars's exploding dice progression. (For no reason, since simple algebra did it more directly.)
But Riemann is my favorite mathematician, because in 1851 he's lecturing about how these equations could be used to rotate three-dimensional space through an imaginary fourth dimension, but of course it's just a folly since rotating through the fourth dimension has no practical application.
Ten years later, up shows James Maxwell saying I don't know what the deal is, but it looks like those "bullshit extra dimensions" equations perfectly describe the relationship between electricity and magnetism... and everyone in the world's MIND is BLOWN. Out of nowhere we get the occultist paradimensional obsession of the late 19th and early 20th centuries, which hits its peak with Rasputin and Houdini, but its purest expression in Lovecraft's stories about fourth-dimensional lifeforms.
And as if that's not enough, Einstein shows up in 1906 with some new thoughts on electromagnetic propagation through curved space, the world goes crazy again... not realizing that his fancy equations are the exact same ones that Riemann presented in 1854, again having told everyone that they're mathematically fascinating but there's no possible practical use for them. Next thing you know, mushroom clouds are going up over the Japanese countryside.
The internet wants you to believe that Tesla was the father of the twentieth century. Tesla can suck it! Riemann gave us Cthulhu AND nukes.
Zupponn wrote:What do you mean? Einstein got all the ladies.
Tzan wrote:Einstein created a mathematical proof which indicated that he could create a gravitational milkshake that "brings all the girls to the yard".
The Shadowscythe wrote:I sexually identify as an Apache gunship.
And my spirit animal is a Marine.
stubby wrote:Zupponn wrote:What do you mean? Einstein got all the ladies.
Einstein wasn't a math prodigy. He was a special relativity prodigy. Ladies had no defense against his special relations.
Apollyon wrote:Is there anything you are not awesome at?
samuelzz10 wrote:Stubby, you know a lot about chicks. How can I start dating one, and have sex later?
The Shadowscythe wrote:I sexually identify as an Apache gunship.
And my spirit animal is a Marine.
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