56.设f(z)=a0+a1z+...+anz^n,(n>=2)满足∑(2,n)i|ai|<=|a1|,证明f(z)在D(0,1)内单叶.

57.a>e,证明方程e^z=az^n在|z|<1内恰有n个根.

58.设D为包含∞的无界域,jordan曲线γ的外部属于D,函数f(z),g(z)在D内亚纯,在γ上满足|g(z)|<|f(z)|<+inf,证明f(z)和f(z)+g(z)在γ外部零点个数减极点个数相同.

59.|ak|<1,f(z)=∏(1,n)(z-ak)/(1-ak'z),证明(1)|b|<1时,f(z)=b在|z|<1内有n个根.(2)|b|>1时,f(z)=b在|z|>1内有n个根.

60.ρ<1,对充分大的n,多项式Pn(z)=1+2z+3z^2+...+nz^(n-1)在|z|<ρ内无零点.

61.记Pn(z)=1+z+z^2/2!+...+z^n/n!.证明对任意给定的R>0,存在N,n>N时,Pn(z)在|z|<R内无零点.

62.计算下述积分
(1)∫(0,2π)dθ/(2-sinθ)
(2)∫(0,+∞)(cosx-e^(-x))/xdx.
(3)∫(0,+∞)lnx/(x^2-1)dx.

63.设Γ是D内可求长jordan曲线且其内部属于D,f(z)和g(z)在D内解析,在Γ上满足|f(z)+g(z)|<|f(z)|+|g(z)|,则f(z)和g(z)在Γ内部有相同零点个数.