# Analytic and plurisubharmonic functions in finite and by Michel Herve PDF

By Michel Herve

ISBN-10: 3540054723

ISBN-13: 9783540054726

Read or Download Analytic and plurisubharmonic functions in finite and infinite dimensional spaces. Course given at the University of Maryland, Spring 1970 PDF

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Additional info for Analytic and plurisubharmonic functions in finite and infinite dimensional spaces. Course given at the University of Maryland, Spring 1970

Sample text

Find its centre and radius. 1 Differentiability and continuity For a real function f (x) of a real variable x the derivative f (x) is deﬁned as the limit f (x + h) − f (x) . 1) f (x + h) − f (x) h is the gradient of the line P Q which converges to the tangent at P as Q → P . So f (x) is the gradient of the tangent at P . For example, if f (x) = x 2 then we have f (x + h) − f (x) (x + h)2 − x 2 x 2 + 2xh + h2 − x 2 = = h h h 2 2xh + h = 2x + h, = h which → 2x as h → 0. Therefore f (x) = 2x. 2 Similarly one can in principle go through all the elementary functions of calculus and show they have the derivatives they are supposed to have.

Z2 The residue of e1/z at z = 0 is 1. On the other hand, g(z) = ez /z4 has a pole of order 4 at z = 0, since the Laurent expansion at z = 0 is 1 ez = 4 z z4 = 1+z+ z3 z2 + + ··· 2! 3! 1 1 1 1 11 + ··· . + 3+ + 3! z 2! z2 z4 z The residue of ez /z4 at z = 0 is 1/3! = 1/6. 9 Calculation of Laurent expansions We proceed by way of example. Consider the function f (z) = 1 , 1 + z2 which has singularities at z = ±i. We ﬁnd the Laurent expansion at z = i by putting t = z − i and expanding in powers of t.

Examples 1. Verify the Cauchy–Riemann equations for the following functions: z3 , 2. 3. 4. 5. 6. ez , sin z, log z. Verify the Cauchy–Riemann formula for the derivative in each case. Prove |z|2 is differentiable only at z = 0. What is its derivative at this point? Prove f (z) = z¯ (|z|2 − 2) is differentiable only on the unit circle |z| = 1. Verify that f (z) = z¯ 2 for these z. Prove that if f (z) is differentiable for all z and is everywhere real valued then f (z) must be constant. Find the Maclaurin expansion of ez sin z up to terms in z5 (i) by differentiating and putting z = 0, (ii) by multiplying the Maclaurin expansions of ez and sin z together.