复变函数与积分变换中的英文单词和短语
微积分常用英文词汇(分章)
英汉微积分词汇English-Chinese Calculus Vocabulary第一章函数与极限Chapter 1 Function and Limit高等数学higher mathematics集合set元素element子集subset空集empty set并集union交集intersection差集difference of set基本集basic set补集complement set直积direct product笛卡儿积Cartesian product象限quadrant原点origin坐标coordinate轴axisx 轴x-axis整数integer有理数rational number实数real number开区间open interval闭区间closed interval半开区间half open interval有限区间finite interval区间的长度length of an interval无限区间infinite interval领域neighborhood领域的中心center of a neighborhood领域的半径radius of a neighborhood左领域left neighborhood右领域right neighborhood映射mappingX到Y的映射mapping of X onto Y满射surjection单射injection一一映射one-to-one mapping双射bijection算子operator变化transformation函数function逆映射inverse mapping复合映射composite mapping自变量independent variable因变量dependent variable定义域domain函数值value of function函数关系function relation值域range自然定义域natural domain单值函数single valued function多值函数multiple valued function单值分支one-valued branch函数图形graph of a function绝对值absolute value绝对值函数absolute value function符号函数sigh function整数部分integral part阶梯曲线step curve当且仅当if and only if (iff)分段函数piecewise function上界upper bound下界lower bound有界boundedness最小上界least upper bound无界unbounded函数的单调性monotonicity of a function 单调增加的increasing单调减少的decreasing严格递减strictly decreasing严格递增strictly increasing单调函数monotone function函数的奇偶性parity (odevity) of a function 对称symmetry偶函数even function奇函数odd function函数的周期性periodicity of a function周期period周期函数periodic function反函数inverse function直接函数direct function函数的复合composition of function复合函数composite function中间变量intermediate variable函数的运算operation of function基本初等函数basic elementary function初等函数elementary function线性函数linear function常数函数constant function多项式polynomial分段定义函数piecewise defined function阶梯函数step function幂函数power function指数函数exponential function指数exponent自然指数函数natural exponential function对数logarithm对数函数logarithmic function自然对数函数natural logarithm function三角函数trigonometric function正弦函数sine function余弦函数cosine function正切函数tangent function半角公式half-angle formulas反三角函数inverse trigonometric function常数函数constant function双曲线hyperbola双曲函数hyperbolic function双曲正弦hyperbolic sine双曲余弦hyperbolic cosine双曲正切hyperbolic tangent反双曲正弦inverse hyperbolic sine反双曲余弦inverse hyperbolic cosine反双曲正切inverse hyperbolic tangent最优化问题optimization problems不等式inequality极限limit数列sequence of number复利compound interest收敛convergence收敛的convergent收敛于a converge to a发散divergence发散的divergent极限的唯一性uniqueness of limits收敛数列的有界性boundedness of a convergent sequence子列 subsequence函数的极限 limits of functions函数当x 趋于0x 时的极限 limit of functions as x approaches 0x单侧极限 one-sided limit左极限 left limit右极限 right limit单侧极限 one-sided limits渐近线 asymptote水平渐近线 horizontal asymptote分式 fractions商定律 quotient rule无穷小 infinitesimal无穷大 infinity铅直渐近线 vertical asymptote夹逼准则 squeeze rule (Sandwich theorem)单调数列 monotonic sequence高阶无穷小 infinitesimal of higher order低阶无穷小 infinitesimal of lower order同阶无穷小 infinitesimal of the same order等阶无穷小 equivalent infinitesimal多项式的次数 degree of a polynomial三次函数 cubic function函数的连续性 continuity of a function增量 increment函数在0x 连续 the function is continuous at 0x左连续 left continuous / continuous from the left右连续 right continuous / continuous from the right连续性 continuity不连续性 discontinuity连续函数 continuous function函数在区间上连续 function is continuous on an interval不连续点 discontinuity point第一类间断点 discontinuity point of the first kind第二类间断点 discontinuity point of the second kind初等函数的连续性 continuity of the elementary functions定义区间 defined interval最大值 global maximum value (absolute maximum)最小值 global minimum value (absolute minimum)零点定理 the zero point theorem介值定理 intermediate value theorem第二章 导数与微分Chapter 2 Derivative and Differential 速度velocity速率speed平均速度average velocity瞬时速度instantaneous velocity匀速运动uniform motion平均速度average velocity瞬时速度instantaneous velocity圆的切线tangent line of a circle割线secant line切线tangent line位置函数position function导数derivative求导法differentiation可导的derivable可导函数differentiable function光滑曲线smooth curve变化率rate of change函数的变化率问题problem of the change rate of a function导函数derived function导数定义域domain of derivative左导数left-hand derivative右导数right-hand derivative单侧导数one-sided derivatives在闭区间[a, b]上可导is derivable on the closed interval [a, b] 指数函数的导数derivative of exponential function幂函数的导数derivative of power function切线的斜率slope of the tangent line截距interceptsx 截距x-intercept直线的斜截式slope-intercept equation of a line点斜式point-slope form切线方程tangent equation焦点focus角速度angular velocity成本函数cost function边际成本marginal cost逐项求导法differentiation term by term积之导数derivative of a product商之导数derivative of a quotient链式法则chain rule隐函数implicit function显函数explicit function隐函数求导法implicit differentiation加速度acceleration二阶导数second derivative三阶导数third derivative高阶导数nth derivative / higher derivative莱布尼茨公式Leibniz formula对数求导法log- derivative参数parameter参数方程parametric equation相关变化率correlative change rata微分differential微分学differential可微的differentiable函数的微分differential of function自变量的微分differential of independent variable微商differential quotient逼近法approximation用微分逼近approximation by differentials间接测量误差indirect measurement error绝对误差absolute error相对误差relative error第三章微分中值定理与导数的应用Chapter 3 MeanValue Theorems of Differentials and the Application ofDerivatives均值定理mean value theorem罗尔定理Roll’s theorem费马引理Fermat’s lemma拉格朗日中值定理Lagrange’s mean value theorem驻点stationary point稳定点stable point临界点critical point辅助函数auxiliary function拉格朗日中值公式Lagrange’s mean value formula柯西中值定理Cauchy’s mean value theorem洛必达法则L’Hospital’s Rule不定式indeterminate form“0”型不定式indeterminate form of type “”泰勒中值定理Taylor’s mean value theorem 泰勒公式Taylor formula系数coefficient余项remainder term线性近似linear approximation拉格朗日余项Lagrange remainder term麦克劳林公式Maclaurin’s formula佩亚诺余项Peano remainder term阶乘factorial凹凸性concavity上凹（或下凸）concave upward (concave up)下凹（或向上凸的）concave downward (concave down) 拐点inflection point极值extreme value函数的极值extremum of function极大与极小值maximum and minimum values极大值local (relative) maximum最大值global (absolute) maximum极小值local (relative) minimum最小值global (absolute) minimum目标函数objective function收入函数revenue function斜渐进线slant asymptote曲率curvature弧微分arc differential平均曲率average curvature曲率园circle of curvature曲率中心center of curvature曲率半径radius of curvature渐屈线evolute渐伸线involute根的隔离isolation of root隔离区间isolation interval切线法tangent line method第四章不定积分Chapter 4 Indefinite Integrals 原函数primitive function / antiderivative积分integration积分学integral积分号sign of integration被积函数integrand积分变量integral variable积分常数constant of integration积分曲线integral curve积分表table of integrals换元积分法integration by substitution有理代换法rationalizing substitution三角代换法trigonometric substitutions分部积分法integration by parts分部积分公式formula of integration by parts有理函数rational function真分式proper fraction假分式improper fraction部分分式partial fractions三角积分trigonometric integrals第五章定积分Chapter 5 Definite Integrals曲线下方之面积area under a curve曲边梯形trapezoid with曲边curve edge窄矩形narrow rectangle曲边梯形的面积area of trapezoid with curved edge积分下限lower limit of integral积分上限upper limit of integral积分区间integral interval分割partition黎曼和Riemannian sum积分和integral sum可积的integrableSimpson 法则逼近法approximation by Simpson’s rule梯形法则逼近法approximation by trapezoidal rule矩形法rectangle method曲线之间的面积area between curves积分中值定理mean value theorem of integrals函数在区间上的平均值average value of a function on an interval 牛顿－莱布尼茨公式Newton-Leibniz formula微积分基本公式fundamental formula of calculus微积分基本定理fundamental theorem of calculus变量代换change of variable换元公式formula for integration by substitution递推公式recurrence formula反常积分improper integral反常积分发散the improper integral is divergent反常积分收敛the improper integral is convergent无穷限的反常积分improper integral on an infinite interval无界函数的反常积分improper integral of unbounded functions瑕点flaw绝对收敛absolutely convergent第六章定积分的应用Chapter 6 Applications of the Definite Integrals 元素法the element method面积元素element of area平面plane平面图形的面积area of a plane figure直角坐标（又称“笛卡儿坐标”）Cartesian coordinates / rectangular coordinates x 坐标x-coordinate坐标轴coordinate axes极坐标polar coordinates极轴polar axis极点pole圆circle扇形sector抛物线parabola椭圆ellipse椭圆的轴axes of ellipse蚌线conchoid外摆线epicycloid双纽线lemniscate蚶线limacon旋转体solid of revolution, solid of rotation旋转体的面积volume of a solid of rotation旋转椭球体ellipsoid of revolution, ellipsoid of rotation曲线的弧长arc length of a curve可求长的rectifiable光滑smooth功work水压力water pressure引力gravitation变力variable force第七章空间解析几何与向量代数Chapter 7 Space Analytic Geometry and Vector Algebra纯量（标量）scalar向量vector自由向量free vector单位向量unit vector零向量zero vector相等equal平行parallel平行线parallel lines向量的线性运算linear operation of vector加法addition减法subtraction数乘运算scalar multiplication三角形法则triangle rule平行四边形法则parallelogram rule交换律commutative law结合律associative law分配律distributive law负向量negative vector三角不等式triangle inequality对角线diagonal差difference余弦定理（定律）law of cosines空间space空间直角坐标系space rectangular coordinates坐标平面coordinate plane卦限octant向量的模modulus of vector定比分点definite proportion and separated point中点公式midpoint formula等腰三角形isosceles triangle向量a与b的夹角angle between vector a and b方向余弦direction cosine方向角direction angle投影projection向量在轴上的投影projection of a vector onto an axis向量的分量components of a vector对称点symmetric point数量积（点积，内积）scalar product (dot product, inner product)叉积（向量积，外积）cross product (vector product, exterior product) 混合积mixed product锐角acute angle流体fluid刚体rigid body角速度angular velocity平行六面体parallelepiped平面的点法式方程point-norm form equation of a plane法向量normal vector平面的一般方程general form equation of a plane三元一次方程three-variable linear equation平面的截距式方程intercept form equation of a plane两平面的夹角angle between two planes点到平面的距离distance from a point to a plane空间直线的一般方程general equation of a line in space方向向量direction vector直线的点向式方程point-direction form equations of a line直线的对称式方程symmetric form equation of a line方向数direction number直线的参数方程parametric equations of a line两直线的夹角angle between two lines垂直perpendicular垂直线perpendicular lines直线与平面的夹角angle between a line and a planes 平面束pencil of planes平面束的方程equation of a pencil of planes行列式determinant系数行列式coefficient determinant曲面方程equation for a surface球面sphere球体spheroid球心ball center轨迹方程locus equation旋转轴rotation axis旋转曲面surface of revolution母线generating line圆锥面cone顶点vertex半顶角semi-vertical angle旋转双曲面revolution hyperboloids旋转单叶双曲面revolution hyperboloids of one sheet 旋转双叶双曲面revolution hyperboloids of two sheets 柱面cylindrical surface, cylinder圆柱circular cylinder圆柱面cylindrical surface准线directrix抛物柱面parabolic cylinder二次曲面quadric surface截痕法method of cut-off mark椭圆锥面elliptic cone椭球面ellipsoid双曲面hyperboloid单叶双曲面hyperboloid of one sheet双叶双曲面hyperboloid of two sheets旋转椭球面ellipsoid of revolution抛物面paraboloid椭圆体ellipsoid椭圆抛物面elliptic paraboloid旋转抛物面paraboloid of revolution双曲抛物面hyperbolic paraboloid马鞍面saddle surface椭圆柱面elliptic cylinder双曲柱面hyperbolic cylinder抛物柱面parabolic cylinder空间曲线space curve交线intersection curve空间曲线的一般方程general form equations of a space curve空间曲线的参数方程parametric equations of a space curve螺线spiral / helix螺矩pitch投影柱面projecting cylinder第八章多元函数微分法及其应用Chapter 8 Differentiation of Functions of Several Variables and Its Application 一元函数function of one variable二元函数binary function邻域neighborhood去心邻域noncentral neighborhood方邻域square neighborhood圆邻域circular neighborhood内点interior point外点exterior point边界点frontier point, boundary point聚点point of accumulation导集derived set开集open set闭集closed set连通集connected set开区域open region闭区域closed region有界集bounded set无界集unbounded setn维空间n-dimensional space多元函数function of several variables二重极限double limit多元函数的连续性continuity of function of several variables连续函数continuous function不连续点discontinuity point一致连续uniformly continuous偏导数partial derivative对自变量x的偏导数partial derivative with respect to independent variable x高阶偏导数partial derivative of higher order二阶偏导数second order partial derivative混合偏导数hybrid partial derivative全微分total differential偏增量partial increment偏微分partial differential全增量total increment可微分differentiable必要条件necessary condition充分条件sufficient condition叠加原理superposition principle全导数total derivative中间变量intermediate variable隐函数存在定理theorem of the existence of implicit function 光滑曲面smooth surface曲线的切向量tangent vector of a curve法平面normal plane向量方程vector equation向量值函数vector-valued function切平面tangent plane法线normal line方向导数directional derivative等高线level curve梯度gradient数量场scalar field梯度场gradient field向量场vector field势场potential field引力场gravitational field引力势gravitational potential曲面在一点的切平面tangent plane to a surface at a point曲线在一点的法线normal line to a surface at a point无条件极值unconditional extreme values鞍点saddle point条件极值conditional extreme values拉格朗日乘数法Lagrange multiplier method拉格朗日乘子Lagrange multiplier经验公式empirical formula最小二乘法method of least squares均方误差mean square error第九章重积分Chapter 9 Multiple Integrals重积分multiple integrals二重积分double integral可加性additivity累次（逐次）积分iterated integral体积元素volume element二重积分变量代换法change of variable in double integral极坐标表示的面积area in polar coordinates扇形的面积area of a sector of a circle极坐标二重积分double integral in polar coordinates三重积分triple integral直角坐标系中的体积元素volume element in rectangular coordinate system 柱面坐标cylindrical coordinates柱面坐标系中的体积元素volume element in cylindrical coordinate system 球面坐标spherical coordinates球面坐标系中的体积元素volume element in spherical coordinate system 剥壳法shell method圆盘法disk method反常二重积分improper double integral曲面的面积area of a surface质心center of mass静矩static moment密度density形心centroid转动惯量moment of inertia参变量parametric variable第十章曲线积分与曲面积分Chapter 10 Line (Curve) Integrals and Surface Integrals对弧长的曲线积分line integrals with respect to arc length第一类曲线积分line integrals of the first type对坐标的曲线积分line integrals with respect to x, y, and z第二类曲线积分line integrals of the second type有向曲线弧directed arc单连通区域simple connected region复连通区域complex connected region路径无关path independence格林公式Green formula顺时针方向clockwise逆时针方向counterclockwise区域边界的正向positive direction of region boundary第一类曲面积分surface integrals of the first type旋转曲面的面积area of a surface of a revolution对面积的曲面积分surface integrals with respect to area有向曲面directed surface对坐标的曲面积分surface integrals with respect to coordinate elements第二类曲面积分surface integrals of the second type有向曲面之元素element of directed surface高斯公式gauss formula拉普拉斯算子Laplace operator拉普拉斯变换Laplace transform格林第一公式Green’s first formula通量flux散度divergence斯托克斯公式Stokes formula环流量circulation旋度rotation (curl)第十一章无穷级数Chapter 11 Infinite Series一般项general term部分和partial sum收敛级数convergent series余项remainder term等比级数geometric series几何级数geometric series公比common ratio调和级数harmonic series柯西收敛准则Cauchy convergence criteria, Cauchy criteria for convergence 正项级数series of positive terms达朗贝尔判别法D’Alembert test柯西判别法Cauchy test交错级数alternating series绝对收敛absolutely convergent条件收敛conditionally convergent柯西乘积Cauchy product函数项级数series of functions发散点point of divergence收敛点point of convergence收敛域convergence domain和函数sum function幂级数power series幂级数的系数coefficients of power series阿贝尔定理Abel Theorem收敛半径radius of convergence收敛区间interval of convergence幂级数的导数derivative of power series泰勒级数Taylor series麦克劳林级数Maclaurin series二项展开式binomial expansion近似计算approximate calculation舍入误差round-off error (rounding error)欧拉公式Euler’s formula魏尔斯特拉斯判别法Weierstrass test三角级数trigonometric series振幅amplitude角频率angular frequency初相initial phase矩形波square wave谐波分析harmonic analysis直流分量direct component基波fundamental wave二次谐波second harmonic三角函数系trigonometric function system傅立叶系数Fourier coefficient傅立叶级数Fourier series周期延拓periodic prolongation正弦级数sine series余弦级数cosine series奇延拓odd prolongation偶延拓even prolongation傅立叶级数的复数形式complex form of Fourier series第十二章微分方程Chapter 12 Differential Equation解微分方程solve a differential equation常微分方程ordinary differential equation (ODE)偏微分方程partial differential equation (PDE)微分方程的阶order of a differential equation微分方程的解solution of a differential equation微分方程的通解general solution of a differential equation初始条件initial condition微分方程的特解particular solution of a differential equation初值问题initial value problem微分方程的积分曲线integral curve of a differential equation 可分离变量的微分方程variable separable differential equation 隐式解implicit solution隐式通解implicit general solution衰变系数decay coefficient衰变decay齐次方程homogeneous equation一阶线性方程linear differential equation of first order非齐次non-homogeneous齐次线性方程homogeneous linear equation非齐次线性方程non-homogeneous linear equation常数变易法method of variation of constant暂态电流transient state current稳态电流steady state current伯努利方程Bernoulli equation全微分方程total differential equation积分因子integrating factor高阶微分方程differential equation of higher order悬链线catenary高阶线性微分方程linear differential equation of higher order自由振动的微分方程differential equation of free vibration强迫振动的微分方程differential equation of forced oscillation串联电路的振荡方程oscillation equation of series circuit二阶线性微分方程second order linear differential equation线性相关linearly dependence线性无关linearly independence二阶常系数齐次线性微分方程second order homogeneous linear differential equation with constant coefficient二阶变系数齐次线性微分方程second order homogeneous linear differential equation with variable coefficient特征方程characteristic equation无阻尼自由振动的微分方程differential equation of free vibration with zero damping固有频率natural frequency简谐振动simple harmonic oscillation, simple harmonic vibration微分算子differential operator待定系数法method of undetermined coefficient共振现象resonance phenomenon欧拉方程Euler equation幂级数解法power series solution数值解法numerical solution勒让德方程Legendre equation微分方程组system of differential equations常系数线性微分方程组system of linear differential equations with constant coefficient。
复变函数与积分变换知识点总复习
解析函数 f (z) 的导数仍为解析函数, 它的 n阶
导数为:
f
(n)
( z0
)
n! 2πi
C
(z
f
(z) z0 )n1
dz
(n 1,2,)
其中C 为在函数 f (z) 的解析区域 D内围绕 z0 的
任何一条正向简单闭曲线, 而且它的内部全含于 D.
8.调和函数与解析函数的关系
调和函数
满足 Laplace
但u iv不是解析函数。
证明：
因为 u x
2x,
2u x 2
2,
u y
2 y,
2u y 2
2,
2u 2u 2 2 0,所以，u是调和函数。 x2 y2
同理 2v 6x2 y 2y3 , 2v 6x2 y 2y3 , x2 (x2 y2 )3 y2 (x2 y2 )3
2v x 2
解：u(x, y) a ln(x2 y2 ),v(x, y) arct an y ,则 x
u 2ax , u 2ay , v y , v x , x x2 y2 y x2 y2 x x2 y2 y x2 y2 在区域x 0内连续，且 u v , v u 在区域x 0上成立时，2a 1, x y x y 即，当a 1 时，函数f (z)在区域x 0内是解析的。
Байду номын сангаас
而 u y2, u 2xy, v 2xy, v x2,在复平面上
x
y
x
y
处处连续，当x y 0时满足C R方程，
故f (z)仅在(0,0)点可导，在复平面上处处不解析。
2）因为f (z) x2 iy,则u(x, y) x2, v(x, y) y,
Chapter 1 复变函数与积分变换(英文版)
Polar
representation
of
complex
numbers
simplifies the task of describing geometrically the
product of two complex numbers. Let z1 r1 (cos1 isin 1 ) and z2 r2 (cos 2 isin 2 ) .
As a result of the preceding discussion, the second equality in Th3 should be written as arg z1z2 arg z1 arg z2 (mod 2 ) . “ mod 2 ” meaning that the left and right sides of the equation agree after addition of a multiple of 2 to the right side. Theorem 4. (de Moivre’s Formula). If z r (cos isin ) and n is a positive integer, then z n r n (cos n isin n ) . Theorem 5. Let w be a given (nonzero) complex number with polar representation w r (cos isin ), Then the n th roots of w are given by the n complex numbers
a 0i
to stand for a
. In other words, we are this
复变函数与积分变换重点公式归纳
复变函数与积分变换第一章 复变函数一、复变数和复变函数()()()y x iv y x u z f w ,,+== 二、复变函数的极限与连续极限 A z f z z =→)(lim 0连续 )()(lim 00z f z f z z =→第二章 解析函数一、复变函数),(),()(y x iv y x u z f w +==可导与解析的概念。
二、柯西——黎曼方程掌握利用C-R 方程⎪⎩⎪⎨⎧-==xy yx v u v u 判别复变函数的可导性与解析性。
掌握复变函数的导数：yx y x y y x x v iv iu u v iu y fi iv u x f z f +==-=+-=∂∂=+=∂∂=1)('三、初等函数重点掌握初等函数的计算和复数方程的求解。
1、幂函数与根式函数θθθθθin n n n n n e r n i n r i r z w =+=+==)sin (cos )sin (cos 单值函数nk z i n ner z w π2arg 1+== (k =0、1、2、…、n-1) n 多值函数2、指数函数：)sin (cos y i y e e w xz+==性质：（1）单值.（2）复平面上处处解析，zze e =)'(（3）以i π2为周期 3、对数函数ππk i z k z i z Lnz w 2ln )2(arg ln +=++== （k=0、±1、±2……）性质：（1）多值函数,（2）除原点及负实轴处外解析,（3）在单值解析分枝上:kk z z 1)'(ln =。
4、三角函数：2cos iz iz e e z -+= ie e z iziz 2sin --=性质：（1）单值 （2）复平面上处处解析 （3）周期性 （4）无界5、反三角函数（了解）反正弦函数)1(1sin 2z iz Ln iz Arc w -+== 反余弦函数 )1(1cos 2-+==z z Ln iz Arc w性质与对数函数的性质相同。
复变函数与积分变换专业术语(Chapter 3)
复变函数与积分变换（双语）课专业术语Functions of a complex variable and integral transform terms§3.1 The concept of complex integral==================================================== oriented curve (directing curve): 有向曲线positive direction: 正方向complex integral: 复积分integral: 积分integrable: 可积分的integrand: 被积函数integrate: 求积分integration: (n.)求积分integrability: (n.) 可积性integral sign: 积分号property of linearity: 线性additive property: 可加性line integral: 曲线积分double integral: 二重积分definite integral: 定积分broken line: 折线the integral is independent of path: 积分与路径无关§3.2, §3.3 Cauchy integral theorem and Cauchy integral formula==================================================== Cauchy integral theorem: 柯西积分定理Cauchy-Goursat theorem: 柯西-古萨定理indefinite integral: 不定积分the principle of deformation of path: 闭路变形原理Generalized Cauchy integral theorem: 广义柯西积分定理compound curves: 复合闭路Cauchy integral formula: 柯西积分公式formula of higher derivatives: 高阶导数公式derivatives of all orders: 任意阶导数cube: 三次方recursion: 递推mathematical induction: 数学归纳法inverse proposition: 逆命题Cauchy inequality: 柯西不等式entire function: 整函数Fundamental theorem of algebra: 代数基本定理contradiction: 反证法，矛盾reciprocal: 倒数§3.4 analytic function and harmonic function==================================================== harmonic: 调和的harmonic function: 调和函数partial differential equation: (PDE) 偏微分方程Laplace function: 拉普拉斯方程harmonic conjugate: 调和共轭partial integration method: 偏积分法theorem of uniquely determined analytic function: 解析函数的唯一性定理indefinite integration method: 不定积分法。
复变函数与积分变换第7章Fourier变换
为了叙述卷积定理，我们先给出
定义7.7若已知函数f1(t),f2(t)，则积分
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称为函数f1(t)与f2(t)的卷积，记为f1(t)*f2(t)，即
由卷积的定义，容易验证卷积满足下面运算规律
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同理可定义Fourier余弦变换为
其反演公式为
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7.2.2δ函数及其Fourier变换 (1)δ 函数的定义
由Fourier变换的定义可知，f(t)要在(－∞,+∞)上绝对可积，才存在
Fourier变换，这样的条件很强，使许多常见的函数如1,t,sint等都不能进 行Fourier变换。 例7.3设某一电路中原来的电流为0，某一瞬时（设t=0时）进入一单位电量 的脉冲，求电路上的电流i(t).
则
这个定理的证明要用到较多的基础理论，这里从略。
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7.2Fourier变换
7.2.1Fourier变换的概念
定义7.1设函数f(t)满足Fourier积分定理的条件，记
称函数F(ω )为f(t)的Fourier 变换，记为 F(ω )=F［f(t)］，由式（7.7）
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第7章 Fourier变换
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所谓积分变换（Fourier变换、Laplace变换等），就是通过积分运算
8复变函数与积分变换中英文简介
复变函数与积分变换课程编码：08N1120310课程中文名称：复变函数与积分变换课程英文名称：Complex Functions and Integral Transformation总学时：46学分：3.0先修课程：工科数学分析课程简介：主要讲述复变函数与积分变换的基本理论、基本方法及其应用。
复变函数部分包括:1.复数与复变函数;2.解析函数;3.复变函数的积分：包括复变函数的积分、柯西积分定理和柯西积分公式;4.级数：包括幂级数、泰勒级数和罗伦级数;5.留数及其应用;6.保形映射。
积分变换包括:1.傅里叶积分变换;2.拉普拉斯积分变换。
Course Description:There are two parts in this course.The first part is on complex functions:1. Complex Numbers and Complex Functions;2. Analytic Functions;3. Complex Integral: The Integral of complex functions ,Cauchy Integral Theorem and Cauchy Integral Formula;4. Series: Power series Taylor series and Laurent series;5. Residues and application of residues;6. Conformal mappings.The second part is on integral transforms:1. Fourier transforms;2. Laplace transforms.。
复变函数与积分变换知识点
复变函数与积分变换知识点一、复变函数的基本概念与性质：1. 复数及复平面：复数是由实数部分和虚数部分组成的数，通常表示为a+bi，其中i为虚数单位。
复平面是将复数与二维平面上的点一一对应的方法表示复数。
2. 复变函数的定义：复变函数是将复数域上的数映射到复数域上的函数。
通常表示为f(z)=u(x,y)+iv(x,y)，其中u(x,y)和v(x,y)分别为实部函数和虚部函数。
3. 复变函数的导数与解析函数：对于复变函数f(z)=u(x,y)+iv(x,y)，若存在导数f'(z)，则称f(z)在z处可导。
若f'(z)在复平面上处处可导，则称f(z)为解析函数。
4.柯西-黎曼方程：柯西-黎曼方程是解析函数的充分必要条件，即u(x,y)和v(x,y)满足柯西-黎曼方程的偏微分方程组。
5.全纯函数与亚纯函数：全纯函数是指在区域上处处可导的函数，亚纯函数是指在其定义域上除有限个孤立点外处处为全纯函数。
二、积分变换的基本概念与性质：1.积分变换的定义：积分变换是将函数f(t)变换为函数F(s)的方法，表示为F(s)=L[f(t)]，其中L为积分变换算符。
常见的积分变换有拉普拉斯变换和傅里叶变换等。
2. 拉普拉斯变换：拉普拉斯变换是将函数f(t)变换为复变函数F(s)的变换方法，定义为F(s)=∫[0,∞)e^(-st)f(t)dt。
拉普拉斯变换有一系列性质，如线性性、平移性、尺度变换等。
3. 傅里叶变换：傅里叶变换是将函数f(t)变换为复变函数F(ω)的变换方法，定义为F(ω)=∫(-∞,+∞)e^(-iωt)f(t)dt。
傅里叶变换也具有一系列性质，如线性性、平移性、尺度变换等。
4. 反变换：反变换是将复变函数F(s)逆变换为函数f(t)的方法。
对于拉普拉斯变换，反变换为f(t)=1/2πi∫(σ-i∞,σ+i∞)F(s)e^(st)ds；对于傅里叶变换，反变换为f(t)=1/2π∫(-∞,+∞)F(ω)e^(iωt)dω。
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single-valued function
多值函数
multi-valued function
连续
continuity
不等式
inequality
第二章
Chapter 2Analytic Functions
微分
differential
奇点
singularity
解析函数
analytic function
Chapter8Laplace Transforms
拉普拉斯变换
Laplace transform
像
image
柯西积分公式
Cauchyintegral formula
柯西不等式
Cauchyinequality
第四章
Chapter 4Series Expressions of Analytic Functions
复函数序列
sequences of complex function
级数
series
幂级数
power series
纯虚数
pure imaginary number
共轭复数
complex conjugate number
运算
operation
减法
subtraction
乘法
multiplication
除法
division
复平面
complex plane
分配律
distribute rule
交换律
exchange rule
Functions of Complex Variable and Integral Transforms
第一章
Chapter 1Complex Numbers and Functions of Complex Varialble
复数
complex number
实部
real number
虚部
imaginary unit
保角映射
angle-preserving mapping
自映射
self-mapping
不动点
fixed point
分式线性变换
linear fractional transformation
多边形
polygon
第七章
Chapter7Fourier Transforms
傅里叶变换
Fourier transform
本性奇点
essential singularity
极点
pole
m阶极点
pole of order m
当且仅当
if and only if
亚纯函数
meromorphic function
第六章
Chapter6Conformal Mappings
从A到B的转角
oriented angle from a to b
导数
derivative
柯西-黎曼方程
Cauchy-Riemann equation
调和函数
harmonic function
指数函数
exponential function
对数函数
logarithm function
三角函数
trigonometric function
双曲函数
hyperbolic function
函数项级数
series of functions
收敛性
convergence
收敛半径
radius of convergence
泰勒级数
Taylor series
洛朗级数
Laurent series
发散
divergence
麦克劳林级数
Maclaurin series
泰勒级数展开
Taylor series expansion
绝对收敛
absolutely convergent
一致收敛
uniform convergence
部分和
partial sum
第五章
Chapter 5Residues and their Applications
留数
residue
孤立奇点
isolated singularity
可去奇点
removable singularity
复合函数
complex function
复数的三角形式
trigonometrical form of complex number
模
modulus
辐角
argument
乘方
power
开方
extraction
开集
open set
闭集
closed set
邻域
neighborhood
充分必要条件
sufficient and necessary condition
边界点
boundary point
有界集
bounded set
区域
domain
简单闭曲线
simple closed curve
连通区域
connected region
分段光滑
piecewise smooth
无穷远点
point at infinity
复变函数
function of complex variable
幂函数
power function
高阶导数
higher order derivative
求导法则
derivation rule
链式法则
chain rule
定义域
domain
导函数
derivative funion
第三章
Chapter 3Integrals of functions of complex variable
傅里叶积分
Fourier integral
卷积
convolution
线性性
linearity
对称性
symmetry
延迟性
time shifting
积分变换
integral transform
反演公式
inversion formula
共轭傅里叶积分
conjugateFourier integral
广义傅里叶积分
generalizedFourier integral
傅里叶逆变换
inverseFourier transform
傅里叶反演公式
Fourier inversion formula
傅里叶正弦变换
Fourier sine transform
傅里叶余弦变换
Fourier cosine transform
第八章