#noindex '''복소해석학''' 주제: [[복소수,complex_number]] [[복소함수,complex_function]] - curr goto [[함수,function#s-38]] - [[코시-리만_방정식,Cauchy-Riemann_equation]] [[드무아브르_공식,de_Moivre_s_formula]] [[오일러_공식,Euler_formula]] [[일의거듭제곱근,root_of_unity]] [[분지,branch]] [[주치,principal_value]] [[함수,function#s-32]](다변수함수 multivariable function) [[경로적분,contour_integral]] path_independence, independence_of_the_path [[미적분학의기본정리,FTC]]와 관련 [[경로,path]] [[경로,contour]] [[독립,independence]] [[코시-구르사_정리,Cauchy-Goursat_theorem]] [[유수,residue]] { [[로랑_급수,Laurent_series]] 중 가장 중요한 계수 $a_{-1}$ ?? see [[유수정리,residue_theorem]] [[WpKo:유수_(복소해석학)]] [[WpEn:Residue_(complex_analysis)]] https://everything2.com/title/residue Up: [[복소해석,complex_analysis]] } [[유수정리,residue_theorem]] { [[유수,residue]] [[https://terms.naver.com/entry.naver?docId=5669300&cid=60207&categoryId=60207 수학백과: 유수정리]] https://everything2.com/title/Residue+Theorem Up: [[복소해석,complex_analysis]] } [[근방,neighborhood]] [[극점,pole]] - writing [[특이점,singular_point]] { '''특이점: singular point, singularity''' // from wpko { f가 a를 제외한 f의 한 근방에서 해석적이면 a를 f의 고립특이점(isolated singularity)이라 한다. 고립특이점은 다시 다음으로 구분된다. 제거가능특이점 removable singularity [[극점,pole]] 본질적특이점 essential singularity 점 a가 함수 f의 고립특이점이면, f는 a를 제외한 a 근방에서 [[로랑_급수,Laurent_series]] $f(z)=\sum_{n=1}^{\infty}\frac{b_n}{(z-a)^n}+\sum_{n=0}^{\infty}a_n(z-a)^n$ 으로 전개할 수 있는데 처음 합을 주부(principal part) 두번째 합을 해석부(analytic part)라 한다. 로랑 급수에서 주부의 항이 전혀 나타나지 않으면 제거가능 특이점, 유한개만 나타나면 극점, 무한히 많이 나타나면 본질적 특이점. } Sub: [[고립특이점,isolated_singular_point]] { https://mathworld.wolfram.com/IsolatedSingularity.html [[WpKo:고립_특이점]] [[WpEn:Isolated_singularity]] https://encyclopediaofmath.org/wiki/Isolated_singular_point https://everything2.com/title/isolated+singularity } removable_singular_point removable_singularity 제거가능특이점 없앨수있는특이점 https://mathworld.wolfram.com/RemovableSingularity.html https://planetmath.org/removablesingularity [[WpKo:없앨_수_있는_특이점]] [[WpEn:Removable_singularity]] https://encyclopediaofmath.org/wiki/Removable_singular_point essential_singular_point essential_singularity https://mathworld.wolfram.com/EssentialSingularity.html https://planetmath.org/essentialsingularity [[WpKo:본질적_특이점]] [[WpEn:Essential_singularity]] https://encyclopediaofmath.org/wiki/Essential_singular_point https://everything2.com/title/essential+singularity 정칙특이점,regular_singular_point { aka regular_singularity ? 푹스 Fuchs Fuchsian equation { https://encyclopediaofmath.org/wiki/Fuchsian_equation } 프로베니우스 Up: regular_point ? { https://mathworld.wolfram.com/RegularPoint.html } Opp: irregular_singular_point ? { 설명은 wpen참조. mklink [[irregular_singularity]]-바로아래쪽. https://encyclopediaofmath.org/wiki/Irregular_singular_point } https://mathworld.wolfram.com/RegularSingularPoint.html [[WpKo:정칙_특이점]] [[WpEn:Regular_singular_point]] https://encyclopediaofmath.org/wiki/Regular_singular_point https://everything2.com/title/regular+singular+point [[http://wiki.mathnt.net/index.php?title=정칙특이점(regular_singular_points)]] } irregular_singularity { https://mathworld.wolfram.com/IrregularSingularity.html } ---- Twins: https://mathworld.wolfram.com/SingularPoint.html https://mathworld.wolfram.com/Singularity.html [[WpKo:특이점_(해석학)]] https://encyclopediaofmath.org/wiki/Singular_point https://everything2.com/title/Singularity Up: [[복소해석,complex_analysis]] [[점,point]] } <> = trig f__n__s와의 관계 = trig functions 관련. ^^∀^^z∈ℂ, z=x+iy, $\sin z=\frac{e^{iz}-e^{-iz}}{2i}$ and $\cos z=\frac{e^{iz}+e^{-iz}}{2}$ 다음과 같은 친숙한 trig identities는 복소수에서도 똑같이 적용됨 $\sin(-z)=-\sin z$ $\cos(-z)=\cos z$ $\cos^2+\sin^2z=1$ $\sin(z_1\pm z_2)=\sin z_1\cos z_2\pm\cos z_1\sin z_2$ $\cos(z_1\pm z_2)=\cos z_1\cos z_2\mp\sin z_1\sin z_2$ $\sin 2z=2\sin z\cos z$ $\cos 2z=\cos^2 z-\sin^2 z$ 전제는 $z=x+iy$ 인 듯? CHK 삼각함수, [[쌍곡선함수,hyperbolic_function]]관련해 $\sin z=\sin x\cosh y+i\cos x\sinh y$ $\cos z=\cos x\cosh y-i\sin x\sinh y$ 여기에 $\cosh^2 y=1+\sinh^2 y$ 를 적용하면 $|\sin z|^2=\sin^2 x+\sinh^2 y$ $|\cos z|^2=\cos^2 x+\sinh^2 y$ (AEM p846-847) sin, cos, sinh, cosh $\sin z=-i\sinh(iz)$ $\cos z=\cosh(iz)$ $\sinh z=-i\sin(iz)$ $\cosh z=\cos(iz)$ $\sinh z=\sinh x\cos y+i\cosh x\sin y$ $\cosh z = \cosh x \cos y + i \sinh x \sin y$ (AEM 848) = tmp = 정칙 holomorphic [[복소함수,complex_function]]가 특정 점 근방에서 미분가능^^differentiable^^(see [[미분가능성,differentiability]])하면 다음 이름이 붙는다고. holomorphic 정칙(적), 복소해석적 holomorphic function 정칙함수 (이상 두개는 kms 번역) 또는 해석적(complex analytic). - [[해석함수,analytic_function]] [[정칙함수,holomorphic_function]] or [[정칙함수,regular_function]] - 작성중. { /// merge: [[해석함수,analytic_function]]에도 있음 [[WpKo:정칙_함수]] 실함수의 미분가능함수+해석함수에 동시에 대응되는 복소함수의 개념이라 함. } ---- CHK z = r e^^iθ^^ means Log z = ln r + iθ. ## Jeffrey AEM 2001 = tmp links ko = https://m.blog.naver.com/cindyvelyn/221751578003 - 복소해석학 카테고리 1번글 = books = Wiki:VisualComplexAnalysis by Wiki:TristanNeedham https://learning.subwiki.org/wiki/Tristan_Needham "Visual Complex Analysis (a book that has been praised by many as being good at giving geometric intuition of complex analysis)" [[ISBN(0198534469)]] ---- Twins: https://ghebook.blogspot.com/2012/08/complex-analysis.html [[WpKo:복소해석학]] [[WpEn:Complex_analysis]] [[Namu:복소해석학]] [[Wiki:ComplexAnalysis]] https://everything2.com/title/complex+analysis ---- Up: [[해석학,analysis]]