复变函数复习

(一)复数的概念1.复数的概念：z x iy =+，,x y 是实数, ()()Re ,Im x z y z ==.21i =-. 注：一般两个复数不比较大小，但其模（为实数）有大小. ①两个复数相等，当且仅当它们的实部与虚部分别相等。

②一个复数等于零，当且仅当它的实部与虚部同时等于零。

③称复数x+iy 和x-iy 互为共轭复数。

2.复数的表示1）模：z=2）幅角：在0z ≠时，矢量与x 轴正向的夹角，记为()Arg z （多值函数）；主值()arg z 是位于[)π2,0中的幅角。

（()Arg z 有无穷个值，()arg z 是复数z 的辐角的主值()Arg z =()arg z +2k π3）()arg z 与arctan y x之间的关系如下： 当0,x > arg arctanyz x=；当0,arg arctan 0,0,arg arctan yy z x x y y z xππ⎧≥=+⎪⎪<⎨⎪<=-⎪⎩； 4）三角表示：)sin (cos z θθi r +=，其中)(r z g A =θ；注：中间一定是“+”号。

（r=|z|）5）指数表示：θi re =z ，其中)(r z g A =θ。

(二) 复数的运算 1.加减法：若111222,z x iy z x iy =+=+，则()()121212z z x x i y y ±=±+±··2.乘除法：1）若111222,z x iy z x iy =+=+，则()()1212122112z z x x y y i x y x y =-++；()()()()112211112121221222222222222222x iy x iy z x iy x x y y y x y x i z x iy x iy x iy x y x y +-++-===+++-++。

复变函数复习重点(一)复数的概念1.复数的概念：z x iy =+，,x y 是实数, ()()Re ,Im x z y z ==.21i =-. 注：一般两个复数不比较大小，但其模（为实数）有大小.2.复数的表示1）模：22zx y =+；2）幅角：在0z ≠时，矢量与x 轴正向的夹角，记为()Arg z （多值函数）；主值()arg z 是位于(,]ππ-中的幅角。

3）()arg z 与arctan y x之间的关系如下：当0,x > arg arctanyz x=；当0,arg arctan 0,0,arg arctan yy z x x y y z xππ⎧≥=+⎪⎪<⎨⎪<=-⎪⎩；4）三角表示：()cos sin z z i θθ=+，其中arg z θ=；注：中间一定是“+”号。

5）指数表示：i z z e θ=，其中arg z θ=。

(二) 复数的运算1.加减法：若111222,z x iy z x iy =+=+，则()()121212z z x x i y y ±=±+±2.乘除法：1）若111222,z x iy z x iy =+=+，则()()1212122112z z x x y y i x y x y =-++；()()()()112211112121221222222222222222x iy x iy z x iy x x y y y x y x i z x iy x iy x iy x y x y +-++-===+++-++。

2）若121122,i i z z e z z e θθ==, 则()121212i z z z z e θθ+=；()121122i z z e z z θθ-=3.乘幂与方根1） 若(cos sin )i z z i z e θθθ=+=，则(cos sin )nnn in z z n i n z e θθθ=+=。

复变函数论（A ）Ⅰ. Cloze Tests （20102=⨯ Points ）1. If nn n n i i z ⎪⎭⎫⎝⎛++⎪⎭⎫ ⎝⎛-=1173，thenlim =+∞→n n z .2. If C denotes the circle centered at 0z positively oriented and n is apositive integer ，then)(10=-⎰Cn dz z z . 3. The radius of convergence of∑∞=++13)123(n n z n nis .4. The singular points of the function )3(cos )(22+=z z zz f are . 5. 0 ,)ex p(s Re 2=⎪⎭⎫⎝⎛n z z , where n is a positive integer.6.=)sin (3z e dzd z. 7. The main argument and the modulus of the number i -1 are . 8. The square roots of i -1 are . 9. The definition of z e is . 10. Log )1(i -= .Ⅱ. True or False Questions (1553=⨯ Points)1. If a function f is analytic at a point 0z ，then it is differentiable at 0z .（ ）2. If a point 0z is a pole of order k of f ，then 0z is a zero of order k off /1.（ ）3. A bounded entire function must be a constant.（ ）4. A function f is analytic a point 000iy x z += if and only if whose real andimaginary parts are differentiable at ),(00y x .（ ）5. If f is continuous on the plane and =+⎰Cdz z f z ))((cos 0 for every simpleclosed path C , then z e z f z 4sin )(+ is an entire function. ( )Ⅲ. Computations (3557=⨯ Points)1. Find⎰=-+1||)2)(12(5z z z zdz.2. Find the value of ⎰⎰==-+228122)1(sin z z z z dzz dz z ze . 3. Let )2)(1()(--=z z zz f ，find the Laurent expansion off on the annulus{}1||0:<<=z z D .4. Given λλλλd z z f C⎰-++=345)(2，where {}3|:|==z z C ，find )1(i f +-'.5. Given )1)(1(sin 1)(2+-+=z z zz f ，find )1),(Res()1),(Res(-+z f z f .Ⅳ. Verifications (30310=⨯ Points)1. Show that if )(0)()(C z z f k ∈∀≡, then )(z f is a polynomial of order k <.2. Show that 012797lim 242=+++⎰+∞→R C R dz z z z , where R C is the circle centered at 0 with radius R .3. Show that the equation 012524=-+-z z z has just two roots in the unite disk复变函数论（B ）Ⅰ. Cloze Tests （20102=⨯ Points ）1. If nn n n i i z ⎪⎭⎫⎝⎛++⎪⎭⎫ ⎝⎛-=1162，thenlim =+∞→n n z .2. If C denotes the circle centered at 0z positively oriented and n is apositive integer ，then)(10=-⎰Cn dz z z . 3. The radius of the power series∑∞=+12)1(n n z nis .4. The singular points of the function )1(sin )(2+=z z zz f are . 5. 0 ,)ex p(s Re 2=⎪⎭⎫⎝⎛n z z , where n is a positive integer.6.=z e dzd z2cos . 7. The main argument and the modulus of the number i -1 are . 8. The square roots of 1+i are . 9. The definition of z cos is . 10. Log )1(i += .Ⅱ. True or False Questions (1553=⨯ Points)1. If a function f is differentiable at a point 0z ，then it is continuous at 0z .（ ）2. If a point 0z is a pole of order m of f ，then 0z is a zero of order m off /1.（ ）3. An entire function which maps the plane into the unite disk must be aconstant.（ ）4. A function f is differentiable at a point 000iy x z += if and only if whosereal and imaginary parts are differentiable at ),(00y x and the CauchyRiemann conditions hold there.（ ）5. If a function f is continuous on the plane and=⎰Cdz z f )(0 for everysimple closed contour C , then z z f sin )( is an entire function. ( )Ⅲ. Computations (3557=⨯ Points)1. Find⎰=-+1||)2)(12(z z z zdz.2. Find the value of ⎰⎰==-+223122)1(sin z z z z dzz dz z ze . 3. Let )2)(1()(--=z z zz f ，find the Laurent expansion off on the annulus{}1||0:<<=z z D .4. Given λλλλd z z f C⎰-++=142)(2，where {}3|:|==z z C ，find )1(i f +-'.5. Given )1)(1(sin )(2+-=z z zz f ，find )1),(Res()1),(Res(-+z f z f .Ⅳ. Verifications (30310=⨯ Points)1. Show that the function iy x e e z z f ---=)2()(2is an entire function.2. Show that if )(0)()(C z z f m ∈∀≡, then )(z f is a polynomial of orderm <.3. Show that 0651lim 242=+++⎰+∞→R C R dz z z z , where R C is the circle centered at 0 with radius R .复变函数论（C ）Ⅰ. Cloze Tests （20102=⨯ Points ）1. If nnn n i i z ⎪⎭⎫⎝⎛++⎪⎭⎫ ⎝⎛+=3131，thenlim =+∞→n n z .2. If C denotes any simple closed contour and 0z is a point inside C , then)(sin 0=-⎰Cn dz z z z, where n is an integer. 3. The radius of convergence of the power series∑∞=-12)63(n n z nis .4. The singular points of the function )2(cos )(244-+=z z z z z f are .5. 0 ,)ex p(s Re =⎪⎭⎫⎝⎛m z z , where m is a positive integer.6. The main argument and the modulus of the number iie 45π are . 7. The integral of the function )(sin )(2ti t t t w += on ]1,1[- is . 8. The definition of z sin is . 9. Log )1(i -= .10. The solutions of the equation 013=-zi e are .Ⅱ. True or False Questions (1553=⨯ Points)1. If a function f is continuous at a point 0z ，thenit is differentiable at 0z .（ ）2. If a point 0z is a pole of order m of f ，then there is a function ϕ that isanalytic at 0z with 0)(0≠z ϕ such that mz z z z f )()()(0-=ϕ on somedeleted neighborhood of 0z .（ ）3. An entire function which is identically zero on a line segment must beidentically zero.（ ）4. A function f is differentiable on open set D if and only if whose real andimaginary parts are differentiable on D and the Cauchy Riemann conditions hold on D .（ ）5. If a function f is continuous on the plane and=⎰Cdz z f )(0 for everysimple closed path C , then 0)(=z f for all z . ( )Ⅲ. Computations (3557=⨯ Points)1. Find⎰=++1||)23)(13(9z z z zdz.2. Find the value of ⎰⎰==-+-222142)1(sin z z z dzz dz z zz . 3. Let )2)(1(3)(2++=z z z z f ，find the Laurent expansion of f on the annulus{}1||0:<<=z z D .4. Given ξξξξd z z f C ⎰-++=543)(2，where {}4|:|==z z C ，find )2(i f +'.5. Find ⎪⎪⎭⎫⎛+i z z ,)1(4Res 222. Ⅳ. Verifications (30310=⨯ Points)1. Show that 0233lim 242=+++⎰+∞→RC R dz z z z , where R C is the circle centered at 0 with radius R .2. Suppose that f is analytic and ||f is a constant on a domain a domainD , prove that a z f =)( for some constant a and all D z ∈.3. Show that the equation z z z z -=+-127234 has just three roots in the unite disk.《复变函数论》试题（D ）Ⅰ. Cloze Tests （20102=⨯ Points ）1. If nnn n i i z ⎪⎭⎫⎝⎛++⎪⎭⎫ ⎝⎛-=1153，then lim =+∞→n n z . 2. If C denotes the circle centered at 0z positively oriented and n is apositive integer ，then)(10=-⎰C n dz z z . 3. The radius of the power series∑∞=++13)12(n n z n nis .4. The singular points of the function )3(cos )(2+=z z zz f are .5. 0 ,)ex p(s Re 2=⎪⎭⎫⎝⎛n z z , where n is a positive integer.6.=)sin (5z e dzd z. 7. The main argument and the modulus of the number i -1 are . 8. The square roots of 1+i are . 9. The definition of z e is . 10. Log )1(i += .Ⅱ. True or False Questions (1553=⨯ Points)1. If a function f is differentiable at a point 0z ，then it is analytic at 0z .（ ）2. If a point 0z is a pole of order k of f ，then 0z is a zero of order k off /1.（ ）3. A bounded entire function must be a constant.（ ）4. A function f is analytic a point 000iy x z += if and only if whose real andimaginary parts are differentiable and the Cauchy Riemann conditions hold in a neighborhood of ),(00y x .（ ）5. If a function f is continuous on the plane and=⎰Cdz z f )(0 for everysimple closed contour C , then z e z f z sin )(+ is an entire function. ( )Ⅲ. Computations (3557=⨯ Points)1. Find⎰=-+1||)2)(12(z z z zdz.2. Find the value of ⎰⎰==-+223122)1(sin z z z z dzz dz z ze . 3. Let )2)(1()(--=z z zz f ，find the Laurent expansion off on the annulus{}1||0:<<=z z D .4. Given λλλλd z z f C⎰-++=142)(2，where {}3|:|==z z C ，find )1(i f +-'.5. Given )1)(1(sin )(2+-=z z zz f ，find )1),(Res()1),(Res(-+z f z f .Ⅳ. Proving (30310=⨯ Points)1. Show that if )(0)()(C z z f m ∈∀≡, then )(z f is a polynomial of order m <.2. Show that 012783lim 242=+++⎰+∞→R C R dz z z z , where R C is the circle centered at 0 with radius R .3. Show that the equation 012524=-+-z z z has just two roots in the unitedisk.《复变函数论》试题（E ）Ⅰ. Cloze Tests （20102=⨯ Points ）1. If nn n i n n z ⎪⎭⎫⎝⎛++-=211，thenlim =+∞→n n z . 2. If C denotes the circle centered at 0z and n is an integer ，then)(1210=-⎰C n dz z z i π. 3. The radius of the power series∑∞=+12)1(n n z nis .4. The singular points of the function 1cos )(2+=z zz f are . 5. 0 ,sin s Re 2=⎪⎭⎫⎝⎛n z z , where n is a positive integer.6.=z e dzd z2sin . 7. The main argument and the modulus of the number i +1 are . 8. The square roots of )0(>A Ai are . 9. The definition of z cos is . 10. Log )22(i += .Ⅱ. True or False Questions (1553=⨯ Points)1. If a function f is differentiable at a point 0z ，then it is continuous at 0z .（ ）2. If a point 0z is a zero of order n of f ，then 0z is a pole of order n off /1.（ ）3. There is a non-constant entire function which maps the plane into the disk1000||<z .（ ）4. A function f is differentiable at a point 000iy x z += if and only if whosereal and imaginary parts are differentiable at ),(00y x and the Cauchy Riemann conditions hold there.（ ）5. If a function f is continuous on the plane and=⎰Cdz z f )(0 for everysimple closed contour C , then it is an entire function. ( )Ⅲ. Computations (3557=⨯ Points)1. Find the integral ⎰+C zdz z e 12, where C is the circle 7||=z .2. Find the value of ⎰⎰==+-+235121)1(sin z z z z dzz dz z ze . 3. Let )2)(1(1)(--=z z z f ，find the Laurent expansion off on the annulus{}1||0:<<=z z D .4. Given λλλλd z z f C⎰-++=765)(2，where {}4|:|==z z C ，find )1(i f +'.5. Given )0(2:,2)(πθθ≤≤=+=i e z C zz z f ，find dz z f C⎰)(.Ⅳ. Proving (30310=⨯ Points)1. Show that 020914lim 242=++-⎰+∞→RC R dz z z z , where R C is the circle centered at 0 with radius R .2. Suppose that f is an entire function and there is a constant M and apositive integer m such that )(|||)(|C ∈∀≤z z M z f m . Prove thatm m z a z a z a z f +++= 221)(for some constants 1a , m a a ,,2 and all z in the plane.3·Show that the equation 01438=-+-z z z has just three roots in the unite disk2005－2006学年第一学期期末考试2003级数学与应用数学专业《复变函数论》试题（C ）Ⅰ. Cloze Tests （20102=⨯ Points ）1. If nnn n i i z ⎪⎭⎫ ⎝⎛++⎪⎭⎫ ⎝⎛+=2121，then lim =+∞→n n z . 2. If C denotes any simple closed contour and 0z is a point inside C , then)(10=-⎰Cn dz z z , where n is an integer. 3. The radius of the power series∑∞=123n n z nis .4. The singular points of the function )2(cos )(24-=z z zz f are .5. 0 ,)ex p(s Re =⎪⎭⎫⎝⎛nz z , where n is a positive integer. 6. The main argument and the modulus of the number iie 42π are . 7. The integral of the function )(sin )(4i t t t w += on ]1,1[- is .8. The definition of z cos is .9. Log )1(i -= .10. The solutions of the equation 012=-zi e are .Ⅱ. True or False Questions (1553=⨯ Points)1. If a function f is continuous at a point 0z ，then it is differentiable at 0z .（ ）2. If a point 0z is a pole of order m of f ，then there is analytic function ϕat 0z with 0)(0≠z ϕ such that m z z z z f )()()(0-=ϕ on some deleted neighborhood of 0z .（ ）3. An entire function which is identically zero on the real axis must be zero.（ ）4. A function f is differentiable on a domain D if and only if whose realand imaginary parts are differentiable on D and the Cauchy Riemann conditions hold on D .（ ）5. If a function f is continuous on the plane and=⎰C dz z f )(0 for everysimple closed contour C , then 0)(=z f for all z . ( )Ⅲ. Computations (3557=⨯ Points)1. Find ⎰=++1||)23)(13(z z z zdz .2. Find the value of ⎰⎰==-+-22216)1(sin z z z dz z dz z z z . 3. Let )2)(1()(2++=z z z z f ，find the Laurent expansion of f on the annulus {}1||0:<<=z z D .4. Given ξξξξd z z f C⎰-++=143)(2，where {}4|:|==z z C ，find )2(i f +'. 5. Evaluate ),)1((Res 222i z z +.Ⅳ. Proving (30310=⨯ Points)1. Show that 02316lim 242=+++⎰+∞→R C R dz z z z , where R C is the circle centered at 0 with radius R .2. Suppose that f is differentiable and ||f is a constant on a domain D , prove that A z f =)( for some constant A and all D z ∈.3. Show that the equation 0127234=-++-z z z z has just three roots in the unite disk.复变函数考试试题（G ）1. 求通过1z 和2z 的线段的参数方程（用复数形式表示）。

复变函数复习题详细答案复变函数复习题详细答案如下：1. 复数的代数形式和几何解释复数 \( z = a + bi \) 可以表示为平面上的一个点 \( (a, b) \)，其中 \( a \) 是实部，\( b \) 是虚部。

复数的模 \( |z| \) 表示该点到原点的距离，即 \( |z| = \sqrt{a^2 + b^2} \)。

2. 复数的运算两个复数 \( z_1 = a + bi \) 和 \( z_2 = c + di \) 的加法和乘法运算如下：\[ z_1 + z_2 = (a + c) + (b + d)i \]\[ z_1 \cdot z_2 = (ac - bd) + (ad + bc)i \]3. 复数的共轭和模复数 \( z = a + bi \) 的共轭为 \( \overline{z} = a - bi \)，模为 \( |z| = \sqrt{a^2 + b^2} \)。

4. 复数的指数形式复数 \( z \) 可以表示为指数形式 \( z = re^{i\theta} \)，其中\( r = |z| \) 是模，\( \theta \) 是 \( z \) 的辐角，满足\( \cos\theta = \frac{a}{r} \) 和 \( \sin\theta = \frac{b}{r} \)。

5. 复数的对数复数 \( z \) 的对数定义为 \( \log z = \log r + i\theta \)，其中 \( r = |z| \)，\( \theta \) 是 \( z \) 的主辐角。

6. 复数的导数设 \( f(z) = u(x, y) + iv(x, y) \) 是复函数，其中 \( z = x +iy \)，则 \( f(z) \) 的导数为：\[ f'(z) = \frac{\partial u}{\partial x} + i\frac{\partialv}{\partial x} \]前提是 \( u \) 和 \( v \) 的偏导数满足柯西-黎曼方程。

一、单选题1.设f(z)=sin z，则下列命题中，不正确的是( )。

A、f(z)在复平面上处处解析B、f(z)以2T为周期C、D、丨f(z)丨是无界的答案： C2.A、iB、-iC、1D、-1答案： B3.下列命题中，不正确的是（）。

A、B、C、若在区域D内有f '(z)=g(z)，则在D内g'(z)存在且解析D、答案： D4.设f(z)在区域D内解析，c为D内任一条正向简单闭曲线，它的内部全属于D．如果f(z)在c上的值为2，那么对c内任一点z0，f(z0)( )A、等于0B、等于1C、等于2D、不能确定答案： C5.下列函数中，为解析函数的是（）。

A、x²-y²-2xyB、x²+xyiC、2(x-1)y+i(y²-x²+2x)D、x³+iy³答案： C6.下列方程所表示的曲线中，不是圆周的为( ).A、B、C、D、答案： B7.函数f(z)在点z可导是f(z)在点z解析的( )A、充分不必要条件B、必要不充分条件C、充分必要条件D、既非充分条件也非必要条件答案： B8.A、2B、2iC、1+iD、2+2i答案： A9.A、不存在的B、唯一的C、纯虚数D、实数答案： D10.A、有界区域B、无界区域C、有界闭区域D、无界闭区域答案： D11.设v(x,y)在区域D内为u(x,y)的共辄调和函数，则下列函数中为D内解析函数的是（）。

A、v(x,y)+iu(x,y)B、v(x,y)-iu(x, y)二、 判断题C 、u(x,y)-iv(x,y)D 、答案： B12.下列数中，为实数的是（ ）。

A 、B 、cos iC 、In iD 、答案： B1.若f (z )在z 0解析，则f (z )在z 0处满足柯西-黎曼条件.A 、正确B 、错误答案： 正确2.若a 是f(z)和g(z)的一个奇点,则a 也是f(z)+g(z)的奇点。